States Effects and Operations: Fundamental Notions of Quantum Theory (Lecture Notes in Physics)

Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M(Jnchen, K. Hepp, Z~rich R. Kippenhahn, MLinchen, H. A. ...

Author: A. Bohm | Karl Kraus

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Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M(Jnchen, K. Hepp, Z~rich R. Kippenhahn, MLinchen, H. A. Weidenmeller, Heidelberg and J. Zittartz, K61n

190 Karl Kraus

States, Effects, and Operations Fundamental Notions of Quantum Theory Lectures in Mathematical Physics at the University of Texas at Austin Edited by A. BShm, J. D. Dollard and W. H. Wootters

Springer-Verlag Berlin Heidelberg New York Tokyo 1983

Author

Karl Kraus Physikalisches Institut der Universit~t WGrzburg Am Hubland, D-8700 WGrzburg Editors

A. B0hm Physics Department, University of Texas Austin, TX 78712, USA J.D. Dollard Mathematics Department, University of Texas Austin, TX 78712, USA W. H. Wootters Department of Physics and Astronomy, Williams College Williamstown, Mass. 01267, USA

ISBN 3-540-12732-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12732-1 Springer-Verlag New York Heidelberg Berlin Tokyo

Library of Congress Cataloging in Publication Data. Kraus, Karl, 1938- States effects, and operations.(Lecture notes in physics; 190) 1. Quantumtheory. I. Title. II. Series.QC174.12.K72 1983 5303'2 83-16906 ISBN 0-387-12732-1 (U.S.) This work is subject to copyright.All rights are reserved,whether the whole or partof the material is concerned,specifically those of translation,reprinting, re-useof illustrations,broadcasting, reproduction by photocopying machineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210

Dem Andenken meiner Mutter

Marie

Kraus

geb.

Paus

4. Juli 1915 - 13. Oktober

1980

Preface

The

lecture notes

of

this volume are based

on a series

of lectures

given in the mathematical physics program at the University of Texas at Austin. of and

These lecture series were started in 1976 with the purpose

establishing of

communication

informing graduate

between mathematicians

students

and

of both departments

physicists

about

recent

developments in mathematics and physics. The lectures are directed at "non-speclallsts",

and

this

volume

also

should

be

suitable

for

a

general audience.

The subject of this volume is the interpretation and the foundations of quantum theory. There exist diverging opinions about this subject. Some people believe that quantum physics is so remote from the usual way of thinking and so foreign to everyday language that its meaning cannot

be communicated

directly.

According

an understanding of quantum mechanics

to them,

the only way to

is to study many separate pro-

blems and thereby acquire an intuitive feeling for the foundations of quantum mechanics.

The

premise

for

the work

the foundations of quantum mechanics

discussed

can be verbalized,

here

is that

and the pre-

sent lectures are an attempt to communicate them.

But

the

champions

concerning

the

of

this field also have

interpretation

of

different

quantum mechanics.

points The

of view

purpose

of

these lecture notes is not to give a comparative review of the different approaches -

this can be found in "Interpretations and Founda-

tions

Theory",

(1981). Ludwig

of

Quantum

Here

just

and his

encompass most

one

school,

H.

approach

Neumann is

BI-Wissenschaftsverlag

presented,

w h i c h appears

of the other orthodox

quantum mechanics.

(Ed.),

to be

the

developed

by

sufficiently general

to

(Copenhagen)

one

interpretations of

Vl

The usual

custom

postulates them

the

a very

usual

operators to only

in

in

in this field

formalism

linear

road

to

with

the

tors

in a Hilbert

this

the

mathematically

fundamental

This

is the road

make

this

Arno B~hm J.D. Dollard W.H. Wootters (Editors)

mathematical quantum

product

more

statements

mechanics

spaces.

of

familiar

rather

in

Even

and

derive

terms

of

quantum

mechanics

objects

of

and

in these lectures

subject

accessible

linear

A more

is to start

operators

them,

from

oneself

procedure.

in terms of these mathematical

fundamental

meaningful

restricting

tedious

than deriving

that has been followed and

language,

is still a very

foundations

space,

important

audience.

of

scalar

one approach,

accessible

the

general

is to start from physically

and

vec-

to formulate quantities. in order to to

a

larger

Author's

The subject

of these notes

fically,

they

structure

of

extent

deal that

when

writing

with

the

standard

also

with

some

these

- at

notes,

in the general

structure

and

the

reader

I had in

already

be

and

physical

the physical

conceptual to this

mechanics,

to specific

More speci-

form and

The

should

quantum

in applications

of

in this

textbooks.

therefore, of

mechanics.

aspects

least

in the usual

formulation

not only

quantum

particular

which

treated

mind

interested

is ordinary

theory,

- are not

Preface

familiar should

problems

interpretation

be but

of the

theory.

As

quantum

mechanics

linear operators matical cal

suitable ding.

are

has

required, if

the same

reason,

full

detail.

in

such

technicalities

These notes Texas

at

Austin

topics

discussed

dings

of

the

on these matters

results

in

an

Spaces"

occasion

several

selfcontalned

readers

will

just

way.

to

in

them

at

who

not

over

including rea-

are presented

too much

them,

the

and

interested concentrate

has

in

given at the University

An

earlier

been

the University

to

rethink

account

published of

1973

at

to

-

statements.

1980.

Marburg

mathemati-

facilitate

deductions

are

"Foundations

aspects, and

to

and

of this mathe-

explain

of lectures

lectures on

of vectors

advanced

order

glance

September

conference

terms

more

most mathematical

in these

lecture

with

-

that

in

Linear

me

tried

on a series

Ordered

ded

have

Whenever

on the more essential

are based

in

some basic knowledge

necessary

I hope

their attention

space,

I

references,

For

formulated

been assumed.

in

of

usually

in Hilbert

framework

tools

is

Quantum [I].

The

of Texas

the whole reformulate

the

in the proceeMechanics invitation

at Austin

subject,

of

to

everything

and to

provi-

extend

my

in a more

VIII

For

this

invitation,

sant stay at Austin, Griffy, ters my

Dr.

for their kind hospitality

English,

I want

W.

I am also very grateful

who read a preliminary version

and

suggested

several

other

to Dr. W.H. Woot-

of my notes,

improvements.

corrected

L a s t but not

to express my hearty thanks to Mrs. Ch. Steinbauer and

Petzold

(WUrzburg),

who

have

task of typewrinting and proofreading, Springer

during my plea-

I want to thank Profs. A. Bohm, J. Dollard, T.A.

and J.A. Wheeler.

(Austin),

least,

and

Verlag

for

his

great

been

burdened

with

the tedious

and to Prof. W. Beiglb~ck from

patience

and

during the preparation of the final typescript.

constant

encouragement

Contents

§I States and Effects §2 Operations

. . . . . . . . . . . . . . . . . . . . .

1

. . . . . . . . . . . . . . . . . . . . . . . . .

13

§3 The First R e p r e s e n t a t i o n T h e o r e m

§4 Composite Systems

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

42

62

§5 The Second R e p r e s e n t a t i o n T h e o r e m

. . . . . . . . . . . . .

81

§6 Coexistent Effects and Observables

. . . . . . . . . . . . .

103

References

. . . . . . . . . . . . . . . . . . . . . . . . . .

150

§i States and Effects

Almost

all treatments

of quantum mechanics

mental

importance

the

physical

to

interpretation

notions

these

different

investigation,

"state"

of these notions,

bly from one author to another. of

of

points

in view

agree

and

however,

funda-

"observable". differs

The

considera-

The discussion of as many as possible

of view is not

of

in ascribing

both

the

the subject

limited

space

of the present

available

and

the

insufficient familiarity of the author with most of them.

We prefer

instead

to follow here as closely as possible one particu-

lar interpretation of quantum mechanics, Ludwig be

and his

school

particularly

theory would We shall

which has been elaborated by

[2], and which the present author considers

satisfactory.

A

still go far beyond

detailed

exposition

the limits

is interested

Ludwig's

of space available here.

try to sketch some of the main ideas only,

any reader who

of

to

in more details

and to encourage

to consult

the original

literature quoted above.

Another

reason

interpretations technical

for

not

entering

a

detailed


parts

of

the

present

discussion

is our conviction

work,

if

suitably

of

that

various the more

reformulated,

are

compatible with most or even all of them. We thus believe that even a reader who disagrees many

profit

completely with the interpretation

from the following

reinterpreting

our

results

in

investigations, terms

of

his

proposed here

even if the burden of

preferred

interpretation

of quantum mechanics is entirely left to himself.

According

to

Ludwig,

any

interpreted "from outside",

physical

theory

is

in

some

sense

to

be

i.e., in terms of "pretheories" not belon-

ging to the theory in question itself. For quantum mechanics in particular, and

these

describe

"pretheories" the

belong

construction

and

to the

realm of classical

application

physics,

of macroscopic

prepa-

2 •

ring

and

measuring

instruments.

advocated

by

who

classical

nature

Bohr, of

A

similar

repreatedly

measuring

point

stressed

instruments

of

the

for

view

has

importance

the

been

of

the

understanding

of

quantum mechanics.

Such

statements

vior

of

should

macroscopic

If

invented. ticular

this

theories

were

Indeed,

be misinterpreted

instruments

in terms of classical namics.

not

true,

can

be

that

quantum

mechanics

of macroscopic

just

would

serves

instruments

the beha-

understood

such as, e.g., mechanics

quantum mechanics

behavior

always

to mean

completely

and electrody-

never

have

to describe

which

been

some par-

can not

be explai-

ned classically.

It

is maintained,

instruments purely

can

during

ter,

are

"naive"

of

quantum

"objective

this

a

Ludwig's ties,

and

respect

events. theory be

radical

changes

theless,

this

cover,

such

changes

as,

objectively

and

always

reference

typical

mechanics

descisive

can

any

e.g.,

real

experimentalists

finally,

to

(at

the

r01e It

in our "naive"

and

are to

application

of

described

in

-

quantum

occurlng

mechanics.

in such instru-

the discharge

e~ents,

always

in

accept

is

of a coun-

much such

the

that

other

occurrences

one

quantum

of

and

interpreted

everyday

concept

concept

has

"things"

as atoms

the

mechanics,

the

with

refined

or electrons.

mechanics ascribing,

creation

of

motivations

of

all its subtle-

consistently

be

of

surrounding

quantum

main

of physical

to

would

in

interpre-

on ascribing

interpretations,

consciousness

fact

of

same

in terms

rests

macroscopic

interpretation

human

entirely

ultimately

the

certain

in

formulated

thus

least)

to

formulated

then

present

with

to show

such

is

events,

disagrees

observable

construction

practice,

without

as

reality"

profoundly

the

and as we all do in everyday life. The physical

instruments

e.g.,

in

"measurements",

in which

in practice,

In

and,

terms,

accepted

way

tation

-

that

in the same spirit,

ments

us.

be

classical

Moreover,

such

however,

without

reality.

such

(Never-

considerably

to

This problem has

also

been

analyzed

by Ludwig,

but

the results

of this deep analysis

cannot even be sketched here.)

According of

to the point

quantum mechanics

of macroscopic preliminary preparing

of view adopted here,

have

instruments

notion

of

thus

instruments.

to be defined

and prescriptions

"state"

the fundamental notions

tells

us

in terms

for their application.

is then most

Experience

operationally

simply given that

in terms

suitably

A of

constructed

instruments can be used to produce ensembles - in principle arbitrarily

large - of

(e.g.,

electrons).

ensemble hand

single microsystems

is then,

notation

for

Ascribing

this

specified

at this lowest the

applied

purpose,

the

notion

the

technical

the

entire preparation

stems

of

are

thus

a

"state"

a

"prestate". of

A

the

accordingly,

prestates,

such

an

We

introduce,

prestate

preparing

is

thus

instrument

is abbreviated here

the same label w shall

the applied preparing procedure - itself.

to

just a short-

procedure.

Such a specification

in different

type considered

level of the theory,

preparation

labels w for prestates;

also be used to denote

particular

like

description

and its mode of application. by using

the

something

for

by

of

instrument - or rather,

Two ensembles

w I $ w2,

of mlcrosy-

if and only

if they

are produced by different preparing procedures.

Another empirical fact is the existence of so called measuring instruments,

which

are

capable

of

undergoing

macroscopically

observable

changes due to ("triggered by") their interaction with single microsystems.

The simplest type of measuring

a single change may be triggered.

instrument

For instance,

is one on which just an originally charged

counter may be found either still charged or discharged,

a~ter it has

been exposed to an electron emitted by some preparing apparatus. result will depend, ter,

and

on whether

loosely or not

speaking, the

on the efficiency

electron

"hits"

it.)

(The

of the coun-

Instruments

of

this type perform so called yes-no measurements:

calling the observa-

ble

usually

change

result

of

the

instrument

of a single measurement

an

"effect",

one

defines

the

to be "yes" if the effect occurs,

and

"no"

if

the

effect

and sometimes rence

of

does

not

occur.

even appropriate,

the effect,

in this way,

and

to associate

vice

the apparatus

It is equally

versa.

(With

then performs

possible,

however,

"yes" with

the non-occur-

its reading

reinterpreted

a different - although close-

ly related - yes-no measurement.)

For

many

values" this

purposes

1 and

0 with

convention,

measurements,

notion

considered nics. can

It be

which

of we

results

also

in

a

The latter the

fit

that

with,

thus

a

broader

e.g.,

class

a movable

instruments

with

them

facts

as

not only

elementary

With of

poin-

more than only two possible

basic

way,

"measured

respectively.

a

- and

need

the

not

quantum

be

mecha-

of any observable

combination

simple

building

-

of

the measurement

standard

are

into

complicated

the

associate

"no",

connected

discuss known

to

and

contains

such more

well

also

"yes"

instruments

in general

first

interpreted,

but

convenient

"observables"

is indeed

measurements.

more

measurements

However,

when

urements,

the

involving

values.

general

is

yes-no

ter on a scale, measured

it

of

prototypes

blocks

for

yes-no

of meas-

more

general

ones. We shall return to this point in §6.

An instrument ratus, As

in

and the

complete

which

shall case

also of

technical

structions

Assume

performing

now then

yes-no measurements be

symbolized

preparing

description

preparing

interacts

the

an or

apparatus

f.

effect

on

assume

such

times.

(Keeping w and f fixed means,

single

identically

the

this

letter,

label

apparatus,

produces apparatus

non-occurrence

Call

experiments,

constructed

w

effect

to

the

occurrence

some

f

usually

stands

including

f.

for

the

a

in-

and its reading.

either

least

the

of

instrument

with

by

instruments,

for its application

a

here

is called an effect appa-

with

this

a

a

single f, of

"single

microsystem,

leading the

in

turn

corresponding

experiment",

and

given w and f, to be repeated of course,

instruments

N

to use the same or at

in all single

experiments.)

We may

then also say that the effect apparatus

f has been applied to

an ensemble of N microsystems in the prestate w.

The

effect

apparatus

experiments, N+ will ment

f

will

not

Nevertheless,

be

determined

large,

large N.

triggered stems

answer

"yes"

in

N+

i.e.,

single

In general,

the outcome of a single experi-

completely

by

the

instruments

w and

f.

in each series of N single experiments with given w and

f the fraction N+/N comes

very

the

and "no" in the remaining N_ = N - N+ cases.

be neither N nor zero;

will

ciently

yield

out roughly

the same,

such that N+/N approaches Thus experience

with

prepared

reproducible

a definite

is suffi-

limit ~(f,w)

tells us that effect apparatuses relative

in a prestate w.

N+/N from ~(f,w)

if only N

frequencies

Deviations

~(f,w)

by

for

f are

mlcrosy-

of the observed values o f

can then be interpreted as statistical errors due to

the finiteness of N.

This

most

laws

of quantum mechanics.

we

call

~(f,w)

apparatus occurs be

crucial

the

the

particular

case

of

the

as

In accordance

w.

for

synonymous

measured

with

just values

basis with

the

Whenever

interpretation

of ~(f,w)

experiments,

empirical

"probability"

in the present

average

is

f in the prestate

understood

single

fact

the

the usual

triggering

statistical terminology,

of

the

of quantum mechanics, frequency",

considered. (yes)

effect

the notion of "probability"

"relative

1

for

Since

and

we may also call ~(f,w)

0

N+/N

it has to as

is

in

the

also

the

(no) o b t a i n e d

in

N

the expectation value of

the f measurement in the prestate w.

Every

experimentalist

preparing

knows

that

some

instrument w or an effect

minor

apparatus

out affecting their statistical behavior, lity

function

intruments

~(f,w).

But

even

two

technical

details

of

a

f may be changed with-

as expressed by the probabi-

completely

different

preparing

w I and w 2 may give rise to the same probabilities

triggering of arbitrary effect apparatuses f; i.e.,

for the

~(f,w I) = ~(f,w2) for all f. Likewise,

there

certainly exist

(slightly,

(1.1)

or even completely,

diffe-

rent) effect apparatuses fl and f2' such that ~(fl,w) = ~(f2,w) for all w; i.e.,

fl and

f2 are

triggered

with

equal

(1.2)

probabilities

in arbitrary

prestates w. If, as usual, we restrict our attention to the probabilities ~(f,w),

as the basic quantities for the formulation of the stati-

stical

of

laws

preparing

instruments

probabilities w2

satisfying

and f,

F

then

inessential.

(1.1),

as

Eqs.

preparing

respectively.

usual,

differences

well

Accordingly,

as

two

(I.i)

and

instruments

All

(1.2)

two

effect

between

instruments

prestates

apparatuses

w

two

these

w I and

fl

and

f2

If thus read as equi-

define

(prestates)

equivalence and

effect

classes W

apparatuses

w or f in a given equivalence

W or F thus behave "statistically", ~(f,w),

any

(1.2), are defined to be equivalent.

relations, of

mechanics,

or effect apparatuses which do not affect

become

satisfying

valence

quantum

class

with respect to the probabilities

in the same way. An equivalence class W is called a state, as whereas - in accordance with Ludwig's

equivalence

class

red

instrument

by

an

ensemble

F an effect. w

in

An ensemble

the

terminology - we call an of N microsystems,

equivalence

class

W,

is

prepa-

called

an

in the state W, whereas an effect apparatus f in the equiva-

lence class F is said to measure the effect F.

By definition of these equivalence classes, rise

to a new function ~(F,W),

the function ~(f,w)

called the probability

gives

for the occur-

rence of the effect F in the state W, as defined by

~(F,W) = 7(f,w) with

f e F

and

w e W.

By

(i.i)

and

depend

on the choice of particular

on

right

the

hand

side

of

(1.3).

(1.2),

(i.3) this

definition

representatives Denoting

the

does not

f of F and w of W

sets of states W and

effects

F

by K and L,

on L × K.

According

respectively,

(1.3)

to its physical

thus

meaning,

defines

this

a function

function

obviously

satisfies 0 ~ ~(F,W) for

all

F e L

and

all

W s K.

~ 1

(1.4)

Moreover,

(1.1) and (1.2) of the equivalence

(1.3)

and

the

definitions

classes W and F also imply that

W 1 = W 2 iff ~(F,W I) = ~(F,W2)

for all F g L ,

(1.5)

F I = F 2 iff ~(FI,W)

for all W e K .

(1.6)

and

This

means,

W 1 @ W2, lead

in

particular,

as well

as

= ~(F2,W)

that

apparatuses

to a different

ensembles

measuring

"statistics",

in

different

and can thus

different

states

effects

F 1 @ F2,

be distinguished

expe-

rimentally.

One

of

the main

mathematical cally

goals

structure

meaningful

function

~.

(The

is no room here the

shall

thus

tions

of

operators

on

state

our

attention

electrons),

as based

may

space

space

of

to

we

the

here

sufficiently

shall also

formalism that

an

"axiomatic"

of

quantum

(ii) projection

approach

quantities

mathematically

Hilbert

space By

systems

possible

H,

in

restricting, (like,

complications

which

and

we

terms we

of

call

moreover,

e.g.,

by density operators W on H, effects,

than

and rela-

which

rules.

operators E on H represent

there

mechanics,

we shall assume - as usual - that

(i) states may be represented

the

physi-

(See [2].) Rather

the basic

"simple"

avoid

and

and second law.) Again

considered.

se could arise from superselection

Accordingly,

such

represented

system

suitable

of

sets K and L and the probability for

separable)

the derivation

from

this in any detail.

be

(complex,

the

on the first

Hilbert

theory

is

mechanics

example

for granted

a

the

for

to discuss

take

approach

quantum

classical

usual

the

of

postulates

is thermodynamics,

deriving

of Ludwig's

single otherwi-

(iii)

the

probability

for

the

occurrence

of

such

an

effect

E

in

a

state W is given by ~(E,W) with tr denoting

For

simplicity

rent

symbols

tors

on H

and

on

reas stand

for

right

(iii)

for

classes

of notation, states

hand

and

the

on

effects

on

on

other

them

the

side

of

(1.7),

the

left

hand

corresponding

,

(1.7)

trace.

we have avoided

and

representing

the

in

the operator

= tr(EW)

the introduction

the

one hand.

W and E side

physical

hand,

of

and

the

in

(i),

Thus,

denote (1.7)

quantities

of diffeopera(ii),

operators, the

same

(i.e.,

whe-

letters

equivalence

of instruments).

Statement

(i)

shall

corresponds

a

assumption,

the

of density

mean,

unique

more

precisely,

density

set K

operators

of

operator,

states

on H,

i.e.,

may

be

that and

to

vice

identified

of non-negative

every

state

versa. with

there

With

the

(and thus)

this

set K(H) Hermitean

operators W with unit trace: W = W* ~ 0 ,

K(H)

is

a subset

of

the

trace

trW = 1 .

class

B(H)I,

(1.8)

consisting

of

all

opera-

tors T on H for which the trace norm

llTil1 : tr[(T*T) I/2] is

finite.

e.g., its

[3],

For

the

Ch.

i, or [4].

Hermitean

part

operators - K(H)

properties

B(H)~,

is convex;

of B(H) 1 -

As a subset

(1.9)

to be

used

later

on - see,

of B(H) I - or alternatively,

consisting

of

all

Hermitean

trace

belongs

state

mixing.

using

N 1 = %N

to

K(H).

class

i.e., with 0 < % < I and Wl, W 2 g K(H),

(I.I0)

W = %W 1 + (I-%)W 2

also

of

Physically,

this

expresses

the

possibility

of by

Assume

an ensemble

of N >> 1 systems

to be prepared

times

a preparing

instrument

N2 =

w I and

(1-%)N

times

another fixed, the

one,

w 2.

defines

prestates

effect

A

prescription

a new wI

apparatus

of

preparation

and f is

w2

with

applied

this

type,

procedure

relative to the

w,

with called

weights

ensemble,

prepared

~,

~ and

wI

and

w2

the mixing

of

1

it will

~.

be

If

triggered

~(f,Wl)N 1 times

by the subensemble

by the instrument

N(f,w2)N 2 times

by the N 2 systems

the probability

for the triggering of f in the mixed prestate w is

prepared

an

by the instrument

Wl, and w 2. Thus

~(f,w) : (~(f,Wl)N 1 + ~(f,w2)N2)/N

: lu(f,wl) + (l-l)~(f,w2) •

This

!

relation

w~ leads

implies

that

replacing

to a w' equivalent

holds true for equivalence

w I and w 2 by equivalent

to w, and that an analogous

in particular,

tors E, the last relation,

relation also

classes:

N(F,W) = I~(F,WI) + (I-%)~(F,W2)

If applied,

w I and

to effects

described

(I.Ii)

.

by projection

opera-

together with (1.7), leads to

tr(EW) = %tr(EW I) + (1-l)tr(EW 2)

= tr[E(~W I + (I-X)W2) ] . As the an

E

is

arbitrary,

projection arbitrary

this

operator

unit

onto

vector

A = W - IW I - (I-%)W 2.

implies

(I.i0).

(Take,

the one-dimensional

f e H.

Then

(1.12)

Since A is linear,

E = I f >< f l,

subspace

spanned

by

(f,Af) = 0, with

we also have

(f,Af) = 0 for

identity

- ((f-g),A(f-g))

+ i((f-ig),A(f-ig))

yields

e.g.,

becomes

vectors f of arbitrary length, and the polarization

4(f,Ag) = ((f+g),A(f+g))

(1.12)

- i((f+ig),A(f+ig))

(f,Ag) = 0 for all f, g s H, i.e., A = 0.)

(1.13)

10

A

state

W

satisfying

interpretation, not

proper

(i.i0)

a mixture

mixtures,

(i. I0)

with

state,

as

W 1%

called

of the states

i.e.,

W2

is thus

and

which

can

be

are

corresponds

of

this

W I and W 2. States

not

0 < X < i,

is well-known,

in view

to a

W which

represented

called

pure

physical

in

the

form

A

pure

states.

one-dimensional

are

projection

operator, W = if >< f l ,

and

is

usually

"state

vector"

represented

by

the

sufficient

satisfactory states

rid

to

only.

procedures of

For

state

for

many

restrict

like

procedures,

unit

f. In this case, Eq.

~(E,W)

Although

even

the

single

ray

{ei~f}

spanned

by

the

reads

it

discussion

instruments,

nevertheless

quantum

as

state

there

is

of

disregarding

since

(1.14)

= (f,Ef)

above-described

mixtures,

with

(1.7)

purposes,

the by

llflf = 1 ,

are

also

which

very

to

pure

states

"artificial" mixing,

not

one

preparation

would

"simple"

not

get

preparation

nevertheless

do

not

pro-

duce pure states.

Statement

(ii)

operator vice

E

versa,

operator modify

above

on H every

are

yes-no As we

assumption

on H as describing which

meant

here

an

effect.

describes

E on H. this

is

measurement shall

by

This

and

imply

Usually

that one

every

also

can be described

however,

admitting

effects.

non-negative

see,

to

there

a larger

set L(H)

(therefore)

assumes

that,

by a projection

are

good

set L(H)

consists

projection

reasons

to

of operators

F

of all operators

F

Hermitean,

and

bounded

from

above by the unit operator:

0 ~ F = F* < 1 .

The ment

set

of

(iii)

projection

operators

is then generalized

(1.15)

E is a proper

to arbitrary

subset

effects

of L(H).

State-

F by requiring

Ii

~(F,W)

(i.i6)

= tr(FW)

for all effects F e L ~ L(H) and all states W ~ K ~ K(H).

The set L(H) is also convex,

i.e., containing with F 1 and F 2 also

(i.i7)

F = ~F i + (i-l)F 2

for

all

real

%

corresponding gous

case of Eq.

"mixed" times

to

the

Since

"mixing"

of

the

projection

I.

an

fl'

and

the effect

still

hope

(1.17)

to projection

shall

try

to

projection

does

that at least

all

not

show

later

on,

"decision

the

convex

- called

subset,

the

is general and them

actual proper

about

tion operators)

to

above

be

as

%N

to be

convex,

such

in the

described

prescription

for

by

a

meas-

so that one might

"simple"

convex of

however,

operators effects"

to be realized set L(H)

states,

of pure

agreement

yield

statements

K(H)

set

that many -

unlikely

set

were

that

f2 has

effect,

the

effect

apparatuses

operators.

effects

the

means

not

if every

sufficiently

cular

respect,

the

is

look very natural,

projection

this

operators

f

fl and f2

Applying

the apparatus

mechanics,

On the contrary,

In

respectively:

be allowed

fied.

over are rather

F2,

times

indeed,

apparatus

as in the analo-

of N microsystems

(I-%)N

not

effect

two effect apparatuses

F 1 and

quantum Now,

particular

F could be obtained,

ensemble

of

would

of

A

by "mixing"

to

operator.

uring

correspond

and

effects

set

effects

formulation

we

f

apparatus

applied.

usual

(1.10),

apparatus the

0

to such an effect

corresponding

As

between

that

which

states.

mixtures

the particular

by Ludwig

of

effects

also

instruments

rSle

is not

justi-

only very

parti-

[2] - which more-

at all in actual

states

rather

hope

describe

than

experiments.

is quite

contains,

But whereas

the pure

preparing

this

similar

as a non-convex

in the latter

case

form only a subset - or perhaps pure

states,

of the decision

in L(H) are much less popular.

to

there

of K(H),

even most

of

corresponding

effects

(projec-

12

Pure states cannot

be

can be characterized prepared

then ask whether distinguished ding

discuss

proper

and how,

physically

to Ludwig

"most

as

sensitive"

this here,

state

mixtures

similarly, from more

[2], decision effects

operationally

in

suitable

effects

above).

One might

F in L(H).

Accor-

are indeed distinguished as the subsets

but will discover another

of decision effects in §6.

(see

the decision effects E could be

general

effects

by the fact that they

of L(H).

We

characteristic

shall not property

13

§2 Operations

We

shall

now

introduce,

cept

of an "operation".

also

be defined

an

preparing state

W.

N >> 1

instrument

w,

so that

moreover,

thus

available

the apparatus

and

for

but

hand,

many

instead the are

"non-destructively"

in

class,

the

stems and

ment w. This

equivalence This

instruments

many the

state,

or

new

con-

this concept

been

shall

theory by

prepared

by

a

is in the corresponding

apparatus is

f is applied

still

"

after

its

f,

which

do

discussion

effect above

present

-

to and

interaction

sense.

with

preparing class

with

the effect apparatus

depends

w,

which

instruments on

both

f constituting

the

which

act

F

the

be

equiva-

f.)

the N microsy-

f as a new ensemble,

state W,

effect then

can

that every effect F

which

the combination

given

To

On

effect

state W as prepared

the

instrument

well

to consider

the apparatus

a quantum

and

of preparing

clearly,

to expect

to considering w

apply.

f in the corresponding

it is allowed

ensemble

as

a given

this

microsystem.

not

f

satisfy

by a suitable apparatus

from the original

instrument

As

not

the

does

apparatuses

instruments

interaction

to this

"destroy"

even reasonable

just amounts

preparing new

microsystem

described,

their

to ascribe

single

each

"non-destructively"

will be different

nal

has

an effect

"absorb"

it appears

situation after

the

used in quantum

the ensemble

experiments

by many different

may be measured

In

first

microsystems

subsequent

there

lence

"effect",

that

that

further

are

instruments

measured

notion,

f.

there

assumption,

and

It was

of

microsystem,

other

"state"

ensemble

each

such

Like

fundamental

[5].

Assume,

(Actually

another

operationally.

Haag and Kastler

Assume

as

by the instruof the origi-

apparatus

belongs

defining

in general

f

to a

the new

as

a

certain state

preparing

instrument

the new preparing

instrument

W.

w and w.

14

We

will

with

show,

another

preparing

however, preparing

instrument

the

new

the

initial

that

state W

inequivalent.

instrument

w'

W)

w'

equivalent

depends

state

combining

on

f and

only.

Accordingly,

To

the

equivalent

to w the

show

there

given

or,

effect to w

apparatus

leads

to a new

in

other

words:

equivalence

class

of w

this,

exists

assume

at least

w

and

that (i.e.,

w'

one effect

f

to

be

apparatus

g, such that

~(g,]) ~ ~(g,]') It

is

the

perfectly

legitimate,

instruments

give

the

gered),

result

however,

f and

g as another

"yes"

("no")

irrespective

to

consider

apparatus

if its "g part"

of the response

in terms of this effect apparatus

of its

the

combination

of

is defined

to

h, which

is triggered "f part".

(not trig-

If reformulated

h, the above inequality

becomes

~(h,w) ~ ~(h,w') , which is a contradiction

since w and w' are equivalent.

Whereas

thus

the new state W depends

ment

w

only

through

for

the

effect

f

within

tuses

its

apparatus the

on the initial

equivalence f does

same

class,

not

hold.

equivalence

the

preparing

analogous

Indeed,

class

statement

different

commonly

yield

instru-

appara-

different

states W, as will be shown later.

An ing

apparatus a given

other

which

ensemble

ensemble

depending Keeping

f applied

on the

-

of N

again

f and

W,

apparatus

it is applied,

tain mapping

in the way

described,

systems

in

some

consisting

of

N

is said f

just

fixed

to perform and

a non-selective

thereby

initial

systems

-

state W into in

a new

a non-selective

varying

the

operation

initial thus

transforman-

state

operation. state

generates

W

to

a cer-

15

: W + W : ~W

(2.1)

of the set K(H) of normalized

density operators

The

all

mapping

~

thus

the given effect physical

describes

apparatus

procedure

possible

(states) into itself.

state

changes

f. As the mathematical

described

above,

this

induced

counterpart

mapping

~

will

by

of the also

be

called here a "non-selective operation".

By using the same effect apparatus

f in a slightly different way, one

can also perform a selective operation,

defined operationally

as fol-

lows. When applied to an ensemble of N >> 1 systems in a state W, the apparatus

f will

after

their

which

have

systems, stems.

be

triggered

interaction triggered

with

f, while

in N+ = ~(f,w)N the

apparatus

disregarding

we arrive at another ensemble, (With

~(f,w) = 0,

N,

an

N+

can

exceptional

also

be

case

to

cases.

f,

the

By

N+

selecting,

microsystems

the remaining

N_ = N - N+

now consisting of N+ microsy-

made be

arbitrarily

considered

large

unless

separately.)

This

^

ensemble

is

also

both applied the

in

a well-defined

instruments,

instruments

w and

new

state W,

depending

again

on

w and f, since as above the combination of

f - now

applied

in a somewhat

different

way,

however - may be considered as a new preparing instrument ~ belonging ^

to a well-defined equivalence class W. ^

As

before,

only,

rather

consider systems cases

the new

state W depends

on

the equivalence

class W of w

than on the particular w chosen from it. To

the application prepared

by

the

of an arbitrary procedure

~.

show this,

effect apparatus g to the N+

Denoting

by N++

the number

of

in which the apparatus g is triggered by this ensemble, we have

by definition ~(g,~) = N++/N+ .

Consider again the combination of the effect apparatuses ^

(2.2)

f and g as a ^

new effect apparatus h, but define now the effect to be measured by h

16

to

occur

if

apparatus

and

are

only

if

triggered.

both

the

Since

f

thus

and

g

used

part

in

a

of

the

different

combined way,

the

^

combination different

of from

discussion riment

of

f and

f

and

g

now

becomes

the apparatus

of combined the

kind

g apparatus

an

effect

h considered

yes-no

above.

measurements

considered,

the

see

either

(For

§6.)

successive

may be interpreted

apparatus

h which

is

a more detailed

In a single

triggering

expe-

of both

the

as a triggering

of g by a

~, or else as a triggering

of h by a

^

system

prepared

by

the procedure

system

prepared

by

the

sponds

to Eq.

instrument

(2.2), whereas

w.

The

the second

first

interpretation

one yields

N++ = ~(h,w)N

.

N+ = ~(f,w)N

,

corre-

the relation

Since

we may rewrite

(2.2)

in the form

~(g,~)

The

right

arbitrary also

hand w'

side

in

of

the

unchanged,

for

= ~(h,w)/~(f,w)

this

same

is

unchanged

equivalence

arbitrary

.

g;

if

class

i.e.,

(2.3)

w

W.

this

is

replaced

Therefore

replacement

by

an

~(g,~)

is

leads

to the

^

same state W.

If ~(f,w)

= 0,

we

use

and

f

of

above

w

does

not

have

N+

according

really

lead

= ~(f,w)N to

the

= 0 as well; selection

to a preparation

i.e.,

the combined

prescription

procedure

described

~. Accordingly,

^

a

"new

state"

W

resulting

and need not - be defined

In

contrast

plied

from

"selective

operation"

can

not

-

in this case.

to a non-selective

to N microsystems

the

operation,

in a state

W,

a

does

selective not

operation,

always

ap-

lead

to N but

The

fraction

^

in

general

N+/N

only

to

N+ < N

is called - quite

systems

legitimately,

in

the

new

of course

state

W.

- the transition

proba-

17 ^

bility

of

exists,

the

and

state

coincides,

the probability it

depends

already

W

the "transition

probability

W,

change

stated

effect for

(If

N+

probability"

= ~(F,W)

the

above

^

W.

by definition

~(f,w) on

+

=

0,

no

is zero.)

final

As this

of the selection

state

transition

procedure,

with

of the effect F in the initial

apparatus

non-selective

f up

to

equivalence

operations,

W

state

only.

however,

As

a corre-

^

sponding

statement

for

the

final

state

W

itself

will

turn

out

to be

wrong.

Mathematically,

a

selective

operation

could

be

identified

with

the

^

mapping before ges,

W ÷ W of

initial

into

for a non-selective

however.

First,

such

final

states

induced

by

it,

as we did

operation.

This would have some disadvanta-

a mapping

would

not

be

defined

on

all

of

^

K(H),

since

mapping

W

does

not

would not specify

re would

have

selective "selective

to

be

operation

exist

operation"

~(f,w)

the transition

given a

if

separately.

slightly in the

= 0.

probabilities, We

different

following

= ~(F,W)

therefore mapping

- which

Second, which

therefo-

associate

~ -

also

is defined,

this

with

a

called

a

for arbi-

trary initial states W ~ K(H), by

= f~(F,W)W

~w

if ~(F,W) ¢ 0 (2.4)

I

0

if ~(F,W) = 0 ^

in

terms

of

the

final

states

W

and

the

transition

probabilities

^

~(F,W).

Since trW = 1 by definition,

~(F,W)

both the transition

probability

= tr(~W)

(2.5)

and - if it exists - the final state ^

W = ~W/tr(~W)

can

then

mapping ~.

be

calculated

for

an

arbitrary

(2.6)

initial

state

W

from

the

18

According K(H)

to its definition

into itself,

thus Hermitean) rators)

with

but rather

~ does not map

the set of states

into the set B(H)~ of non-negative

trace class operators

(or "unnormalized"

tr(~W) ~ I. In view of the additional

tion provided by Eq. a

(2.4),

disadvantage.

(2.5), however,

The

(2.4) is, however,

most

(and

density ope-

physical

informa-

this is an advantage rather than

important

motivation

that ~ is convex-linear,

for

the

definition

i.e., satisfying

~(%W 1 + (I-I)W 2) = I~W 1 + (I-I)~W 2

(2.7)

for arbitrary WI, W 2 e K(H) and all real ~ with 0 < % ~ i.

For

the

proof

of

(2.7),

ties tr(~W I) and tr(~W2)

assume

first

that

the transition

are both non-vanishing.

probabili-

As already discussed

in §I, an ensemble of N systems in the mixed state

W = %W 1 + (l-%)W 2

may

be prepared

the

state

W 2.

(Since

WI,

by applying and

%N = N 1 times an instrument w I preparing

(I-%)N = N 2 times

(2.7)

is

trivially

another

satisfied

instrument

for

~ = 0

w 2 preparing

or

i,

we

also

assume 0 < ~ < i.) Applying the selective operation ~ in this particular

case

then amounts

w2 bY ~I and ~2' apparatus

f

to replacing

respectively,

and

the

the preparing

by adding

above

selection

instruments

w I and

to each of them the effect prescription.

The

state

^

W = ~W/tr(~W)

of

the

ensemble

after

the

operation

~

is

thus

some

mixture, ^^ 1 + (I- f)w 2 ' W^ = %W ^

of

the

states

^

W 1 = ~Wl/tr(~Wl)

and

W 2 = ~W2/tr(~W2)

prepared

by

~)I

and W2' respectively.

The

instrument

surviving

wI

prepares

N1

systems,

NI+ = tr(~Wl)N 1

of

the selection procedure performed with the apparatus

them f, and

19

thus

leaving

the

N2+-- tr(~W2)N 2 NI+ + N2+

preparing

systems.

systems;

instrument

~I"

final

ensemble

The

so the

transition

Similarly, thus

probabilty

of

~2

prepares

contains the state

N+ = change

^

W+Wis tr(~W) = N+/N .

Moreover,

the weight factors in the m i x e d s t a t e

W become

= NI+ = tr(#Wi)Nl = tr(#Wl) N+

tr(~W)N

tr(~W)

and 1

~

tr(~W2)

-

(i

:

-

~)

.

tr(~W) This, finally,

implies ^ ^

~W = tr(~W)W = tr(#W)(%W 1 + (I-~)W 2) ^

= %tr(~Wl)W 1 + (l-%)tr(#W2)W 2 = %~W I +

(I-%)~W 2

,

i.e., Eq. (2.7).

The same result is also obtained in the exceptional cases

tr(#W I) # 0 ,

tr(#W 2) = 0

and tr(~Wl) = tr(~W 2) = 0 .

As

#W 2 > 0 by definition,

case

we

therefore

= (NI/N)(NI+/NI)

get

tr(~W 2) = 0

N2+ = 0,

and

implies thus

~W 2 = 0.

~ = 1

and

= ltr(~Wl) , which implies (2.7):

~W = tr(#W)W 1 : %~W 1 = %#W 1 + (I-I)~W 2 •

In the

first

tr(~W) = NI+/N

20

In the

second

tr(~W)

a

continuing

of

finds

some version

follows: systems

If

we

In

a

effect

shall

in the

E

are,

the

the general

used

is

measured

N+ = tr(EW)N

after

the

one

formula",

here

proper-

discussion

mechanics

reduction

language E

the mathematical

quantum

packet

effect

then

of

trivially.

of

illustrate

textbooks

decision W,

(2.7) holds

investigation

of the "wave

which

in state

the

our

~,

example.

postulate",

red

with

the mapping

specific

tion

~W 1 = ~W 2 = 0 and - since N+ = 0, and thus

= 0 - also ~W = O, so that

Before ties

case we have

by

usually

or "projec-

can be formulated in

systems

an

ensemble

which

measurement,

in

have

the

as

of

N

trigge-

new

state

^

W = EWE/tr(EW).

Since

in

(2.4)

the

leads

case

considered

the

tansition

probability

The corresponding

non-selective

.

(2.8)

operation

is

~W = EWE + (I-E)W(I-E)

ing

tr(EW),

to ~W = EWE

Namely,

is

the

projection

the non-occurrence

N_ = tr((I-E)W)N

postulate of E,

systems

applied

implies

which

have

to

that, not

.

(2.9)

the

effect

after

I-E,

describ-

the measurement,

triggered

the effect

the

E are in

^

the

state

W' = (I-E)W(I-E)/tr((I-E)W).

Therefore,

if

no

selection

is

^

made,

the

state W = ~W after

the E measurement

is a mixture

of W and

^

W'

with

the weight

factors

tr(EW)

and

tr((l-E)W),

respectively,

and

is thus given by (2.9).

It

is

commonly

admitted

that,

in

practice,

there

are

also

E measure-

^

ments

which

(2.9).

do not

For this

lead

reason,

to the final a measurement

late

is sometimes

called

that

such

measurements"

"ideal

an

"ideal are

states

W or W given

satisfying

measurement". only

a

very

by (2.8) and

the projection We shall

postu-

indeed

particular

show

type

of

21

operations,

and

that

actual

not

very

likely

to be of

are

not

decision

effects

anyway. mind

Nevertheless,

as simple

this

type.

Eqs.

the

examples

measurements

decision

Besides

(2.8)

examples

of

and

to effects

(2.9) would

(2.8)

illustrating

this,

and

E

are

F which

not be applicable

(2.9)

the general

effects

should

be kept

discussion,

in

to which

we now return.

By virtue

of

(2.7),

the mapping

unique

way

to a mapping

trace

class

B(H) 1 into

I.e.,

the extended

-

for

¢ : K(H)

÷ B(H)~

simplicity

also

itself,

mapping

which

is

can be extended

denoted

by ¢ - of

complex-linear

and

¢ : B(H) 1 ÷ B(H) I satisfies

for all T, S g B(H) I and all complex

CT ~ 0

(2.10)

(2.18)

and

(2.11)

(2.10)

- or else

below - ~ is also real,

numbers

if

by

a and b; and

T ~ 0 .

its

(2.11)

explicit

construction,

cf.

see

this,

T+ > 0. by may

By

(2.10), be

note

that

(2.11), then,

written

the

any

(2.12)

Hermitean

operators

~T = ~T+ - ~T_ as

Eq.

i.e.,

CT * = (¢T)*

To

the

positive.

¢(aT + bS) = aCT + b¢S

By

in a

~T+ is

T 1 + iT 2 with

T

is

are also

of

the

> 0, and

form

T 1 and

with

thus Hermitean,

Hermitean.

Hermitean

T+ - T

and

A non-Hermitean T2,

so

that,

T

again

by (2.10),

~T* = ¢(T 1 - iT 2) = CT 1 - i~T 2 = (~T 1 + i¢T2)*

Although

the

standard,

we

procedure shall

done in three steps.

of

sketch

extending it here

for

~

from

K(H)

the reader's

to

= (¢T)*

B(H) I

.

is

quite

convenience.

It is

22

In a first of

the

step,

cone

the original

B(H)I

of

mapping

"unnormalized"

which is also convex-linear

~ is extended density

to a mapping

operators

into

(or rather "positive-linear");

~+

itself,

i.e.,

~+(T + T') = #+T + ~+T'

(2.13)

~+aT = a~+T

(2.14)

and

for

all

T, T' s B(H)~

and a l l

numbers

a > 0.

To

achieve

this,

set

~+T = 0 if T = 0, and

(2.15)

~+T = trT.~(T/trT)

if

T ~ 0,

the

last

if 0 # T e B(H)~. T, T' e B(H)~

expression

From this,

(and

neither

being

(2.14) of

them

well-defined follows equal

since

trivially. to

zero,

T/trT g K(H)

Moreover, since

with

otherwise

(2.13) holds trivially), we get T + T' V d~.

trT

= tr(T+T')

T --+ tr(T+T') trT

= ~w

with

V, W = T/trT

+

trT'

T '

tr(T+T') trT'

(l-~)w'

and W' = T'/trT' g K(H)

and 0 < X < I, so that,

by

(2.7) and (2.15),

¢+(T + T') = tr(T+T').%V = tr(T+T')(X~W + (I-X)~W')

-- trT.~W + trT'.~W' = ~+T + ~+T' ,

which proves (2.13).

In a second step, of

the

itself.

real

the mapping

space

B(H) h

If T = T* g B(H)I,

of

~+ is extended to a real-linear mapping Hermitean

trace

we have T = T + -

class

operators

into

T_ with T_+ e B(H)I;

take,

23

e.g., not

T± = (ITi ± T)/2 unique,

with

however;

[TI = (T2) I/2.

e.g.,

with

Such d e c o m p o s i t i o n s

arbitrary

T = (l+c)T+ - (T_ + cT+). Nevertheless,

c > 0

we

also

= S+-

S

with

T+, S± ~ B(H)I,

(2.16)

of T is unique. we

have

have

~r defined by

~r T = ~+T+ - ~+T_

with any of these decompositions

are

For,

if T = T ~ -

T+ + S_ = S+ + T

E B(H)I

T_ and

thus, by (2.13),

~+(T+ + S_) = ~+T+ + ~+S_ = ~+(S+ + T_) = ~+S+ + ~+T_

,

or ~+r+ - ++T_

= ~+S+ - ~+S_

There remains to prove real-linearity of ~r' i.e.,

~r(aT) = a~rT ,

~r(T + T') = ~r r + ~r T'

(2.17)

for all T, T' E B(H)~ and all real a. We have aT = S+ - S_ with

S+ = ±aT± ,

the upper

(lower)

signs

+ S_ = ±aT¥ ~ B(H) I ,

being valid

if a > 0 (a < 0), and thus, with

(2.14) and (2.16),

~r(aT) = ~+S+ - ~+S_ = ± a~+r± ~ a~+r¥

= a(~+T+ - ~+T_) = a~rT

.

Moreover, with the decomposition

T + T' = (T+ + T~) " (r_ + Ti)

we obtain from (2.13) and (2.16)

24

Cr(T + T') = ¢+(T+ + T$) - ¢+(T_ + TJ)

= ¢+T+

Finally, defining,

~r

is

with

- ~+T_ + ¢+T$ -

extended

to

the unique

a

mapping

decomposition

¢+T1 ~c

= Cr T + Cr T'

of

B(H) I into

T = T1 +

•

itself

by

iT 2 of T e B(H) I

into Hermitean T I and T2,

~c T = ~rTl + i~rT 2 •

(2.18)

" ' and using (2.17), we then get With T' = T 1' + IT2,

~c(T + T') = ~c(Tl + T{ + i(r 2 + Ti))

= ~r(Tl + r{) + i~r(r 2 + T~)

= ~rrl + i~rr 2 + ~rrl + i~rT ~

= ~c T + ~c T'

.

(2.19)

Moreover, with real ~ and 8,

(~ + iS)T = (~T 1 - ~T2) + i(~T 2 + BT I) , and thus, again by (2.17),

~c((~ + iB)T) = ~r(~Tl - 8T 2) + i~r(~T 2 + 8T I)

= ~rT1

- 8~rT 2 + i ~ r T 2 + iS~rT 1

= (~ + i~)(¢rT 1 + iCrT 2)

= (~ + iS)~cT .

(2.20)

25

Eqs.

(2.19)

and (2.20) are equivalent

to (2.10),

the complex-lineari-

ty of ~c"

If

T = T*,

(2.18)

yields

~c T E ~r T.

~r T = ~+T if T e B(H)~,

and (2.15)

~+'

are

really

of #c (Eq.

(2.11))

~r

and

~c

Likewise,

implies

(2.16)

implies

~+T = ~T if T c K(H).

extensions

of

the

original

Thus

mapping

: K(H) + B(H)~.

Positivity

#+ on B(H)~, done

in

and ~+T > 0 by

(2.10)

identifying

and

follows (2.15).

(2.11)),

an operation

we

with

immediately

since ~c reduces

As announced

shall

omit

the

to

before

(and already

suffix

in ~c'

the extended mapping

thus

~ : B(H) 1 + B(H)I,

from now on.

We

shall

now prove

that

the mapping

~ is continuous

with

respect

to

the trace norm: IJ~T;II < C HTII1 ,

We recall here the definition

with

some properties

[4].)

Denoting

,

ITi = (T'T)

of the trace norm

by B(H)

Sup tr(~W) < I . W~K(H)

the algebra

1/2

,

11-I[I • (Compare

Jtr(XT)j

0

operators

on

and

if

if

only

Y > 0; moreover, we have

nThus Y > 0

implies

--

010

X > O,

so that,

thus ~ Y > O; involving

i.e.,

with

~ X > O, and

operator

n -

~ is

also

matrices

m-positive.

with

plies t o m a p p i n g s ~ o f B(H) 1 i n t o

Since

if ~ is n-posltive,

--

"ordinary" n > 1 and,

positivity more

so,

~n a n a l o g o u s

"matrix

elements"

argument -

now

from B(H) I - ap-

itself.

is the same as l-positivity, complete

ments. A physically relevant example

positivity,

are

n-positivity

stronger

require-

of a positive but not completely

positive mapping will be discussed in §3.

A mapping ~ : B(H) 1 + B(H) 1 is n-positive mapping ~* that

: B(H) + B(H) is n-positive.

the mapping

(~*)n defined

by

if and only if the adjoint

We show this by first proving

(2.33)

- with

~* for ~ -

is

the

adjoint of the mapping ~n defined by (2.32); i.e.,

(~*)n = (~n)* "

By definition of the adjoint (see Eq. (2.25)),

tr(X.~nT) = tr((~*)nX. ~)

(2.34)

(2.34) means

(2.35)

36

for

arbitrary

since,

X = (Xij) e B(H @ H n)

according

to

"matrix

and T = (Tij) e B(H @ Hn) I.

calculus"

and

Eqs.

(2.32)

and

But

(2.33),

X-~n ~ and (~*)nX. ~ have the matrix representations

X-~nT = (~ Xik'~Tkj) ,

(~*)nX.T = ([ ~*Xik'Tkj) k

k

,

Eqs. (2.31) and (2.25) indeed lead to (2.35):

tr(X.#nT)

=

[ tr(Xik'~Tki ) ik

=

[ tr(~*Xik'Tki)

= tr((~*)nX.T)

•

ik According to (2.34), we may simply write ~

for (~*)n or (~n)*.

Now let ~n be positive. Then #n~ > 0 for all W e K(H 8 Hn) ; therefore, with (2.35), X > 0 implies

tr(~X.W)

for all W, which Vice

versa,

= tr(X.~n ~) > 0

in turn implies

let #~

be

positive.

¢~X > 0, i.e., Then ~ W

positivlty

> 0 and

thus,

of ¢~.

again

by

(2.35), tr(_W,~nT) -- tr(~*W.T) > 0

for all W s K(H ~ Hn) if T e B(H @ Hn)l, T > 0. Therefore ~n ~ > 0; i.e., ~n is positive. This proves the above conjecture.

As already mentioned,

there are physical reasons for postulating that

every

or,

operation

~ -

equivalently,

its adjoint

~* - has

to be

completely positive [7]. As is well known (and as will be elaborated here

in some more

considered

detail

in §4),

the Hilbert

as the state space of a composite

space H @ H

system I +

n

can be

II, consi-

sting of the system I considered up to now, with state space H, and

37

another

microsystem

II

with

state

space

H .

(Since

H

n

sional,

system II is usually

called

is

n-dimen-

n

an n-level

system.)

The mappings

cn considered above then acquire a simple physical interpretation.

Assume

that

ensemble

there

is

no

of N composite

teracting

systems

I

interaction systems

and

II.

between

systems

then consists

Its

state

I

and

of N pairs

is described

II.

An

of non-in-

by some

density

operator W on H @ H . Now apply to system I of each pair the selectin ve operation described by the mapping ~, while leaving system II unaffected;

or,

in other

with a suitable with

system

words:

effect apparatus

let

system

I of

each pair

interact

f (which is supposed not to interact

II), and select those pairs I + II for which the appara-

tus f is triggered. K(H @ Hn) , which

This procedure

can

be

extended

clearly defines

an operation ~ on

to

linear

a

positive

mapping

of

B(H @ Hn) I into itself.

In

particular,

W g K(H) system,

density

and V ~ K(H n)

operators describe

which may be prepared

W

of

the

uncorrelated

form states

by using separate

II

into

pairs

afterwards.

(See

also

=

W @ V

§4.)

with

of the composite

preparing

w for system I and v for system II, and combining I and

W

instruments

the single systems For

such

states

we

must have ! ( w ~ v) = +w ~ v

by the definition with

V = (vii)

,

of the operation ~. On the other hand,

has

the

matrix

representation

relation, cn(W @ V) = ~W @ V ,

follows from (2.32) for the mapping cn" Thus

cn(W @ V) = ~(W @ V)

(vijW),

since W ~ V an

analogous

38

for

all W E K(H)

and all V s K(Hn).

the last equality tors

W @ S

T g B(H)I, latter,

as

n-positive;

Actually

can be extended,

with and

S e B(Hn) ,

finally

an

preceding

§3,

an

then

all

to

all

to

be

first to all opera-

T @ S

T g B(H ~ Hn) I.

has

(separable)

with

Thus

positive.

stronger

that

this

could

be

Hilbert

rather

to postulate

somewhat

however,

argument

"ordinary",

lead us

to be

by linearity,

arbitrary

~n E i;

Therefore

but

the

~

must

be

well

with

an

of Hn,

as

and n was arbitrary.

infinite-dimensional

would

to

operation,

the

describing

Since both ~n and i are linear,

applied

as

space H in place

than

an

n-level

system

II.

This

a positivity property for ~ which appears

than complete apparently

positivity.

stronger

We

property

shall prove is already

in im-

plied by complete positivity.

The effect

apparatus

f used for the operation

perform

another

selective

Namely,

instead

of selecting,

systems

which

the

systems

have

which

operation

triggered have

not

after

~',

~ may also be used to

called

complementary

to

~.

their introduction with f, those

the apparatus triggered

f.

f, we may as well

By

interchanging

select

"yes"

and

"no" in the verbal interpretation of the response of the effect aparatus f, we get another effect apparatus not

triggered,

and vice

versa.

The

f', which is triggered if f is

corresponding

effect

operators

F

and F' are related by

F + F' = 1 ,

since

the occurence

of either F or F'

(2.36)

is certain

by definition,

and

therefore

tr(FW) + tr(F'W) = tr((F + F')W) = 1 = tr(IW)

for all

W e K(H).

F' = 1 - F

(F e L(H)

describing,

in

clearly

this

sense,

implies the

F' e L(H).)

non-occuurrence

The

effect

of F,

is

39

also called complementary

to F - or, briefly,

"not F". By definition,

and F are also complementary to ~' and F', respectively:

&')'

complementarity

=

~

,

(F')'

is thus a symmetric

=

F

;

relation.

Since

the operation

~'

is performed by selecting according to the occurence of F', two operations

~ and ~'

complementary

complementarity

(2.36)

of

to each other are characterized the

corresponding

effects

by the

F = ~*i

and

F' = ~'*i or, equivalently, by

tr(~W) + tr(~'W) = 1

Whereas, F',

by

there

(2.36),

are

for all W e K(H)

an effect F has exactly

in general

many

to a given operation #, since

different

.

(2.37)

one complementary

operations

(2.37) with given

#'

effect

complementary

~ does not uniquely

determine ~'. Both ~ and ~' thus depend on the particular apparatus f used to measure the effect F, rather than on F only.

Finally,

as already

mentioned

at

the beginning

of

this

chapter,

the

effect apparatus f may also be used to perform a non-selective operation.

In this case,

apparatus joint)

f is made

subensembles

no selection with respect - or, of

equivalently,

systems

in

an

sting

initial again

state

of N

W

is

of which

by

the

(by definition complementary

dis-

opera-

Thereby an ensemble of N systems

transformed

systems,

the two

produced

tions ~ and ~' are mixed afterwards.

to the response of the

into

another

N+ = tr(~W)-N

ensemble, are

consi-

in the

state

^

W = #W/tr(~W),

whereas

the

remaining

N

D

= tr(~'W).N

systems

are

in

^

the (Eqs.

state

W' = ~'W/tr(~'W),

(2.5),

according

to

the

physical

meaning

of

(2.6)) and ~'. The final state W resulting from the non^

selective operation is thus a mixture of the states W and W' with the weights N+/N = tr(~W) and N_/N = tr(~'W), respectively; ^

^

= tr(~W)W + tr(#'W)W' = #W + ~'W .

i.e.,

40

Therefore

the mapping

~

: K(H)

+ K(H)

describing

the non-selective

operation (Eq. (2.1)) is simply given by

$=~+~'

(Since

~ and

~'

are

:w

positive,

÷

we have

trW = i. Thus ~ indeed maps K(H) also

be

extended

to a mapping

~=~w+~'w.

W

(2.38)

> 0, while

into itself.)

of

B(H) 1 into

Like

(2.37)

implies

# and #', ~ may

itself.

As

is obvious

from (2.38), this extension is given by

~--~+~'

: T

+

~=~T+~'T

(2.39)

for arbitrary T e B(H) I.

Since

the adjoint ~*

of $ is ~* + ~'*,

the "effect"

corresponding to

= F + F' = 1 .

(2.40)

the nonselective operation ~ is

= $*I = ¢*i + ~ " i

This is also obvious from (and equivalent to)

tr(FW) E tr(~W) = 1

for all W e K(H)

,

which expresses the fact that the transition probability is identically one,

and is thus

"effect"

F = 1

always

gives

effect

thus

selective

describes

the

result

simply

operations

lar "selective"

characteristic

counts

the

"yes".

of a non-selective

trivial An

"yes-no

effect

the systems

measurement"

apparatus

The

which

~ measuring

in any given ensemble.

can thus be considered

ones - the "selection"

operation.

this

As non-

formally as very particu-

being made with respect to the

trivial effect F = I - they need no separate mathematical treatment.

These considerations are illustrated by the example of "ideal measurements"

(Eqs.

(2.8) and (2.9)).

In this case,

mentary to ~ is given by ~'W = E'WE'.

the operation ~' comple-

41

Summarizing operation

the main

has

tion in

the

For

by

at

more

that

on H

least

knows

~

-

position practice.

that

of

or

an

In

a yes-no

a

measurements

by

few -

a

that

measurement "suitable"

such

an

cases

mapping

we

~ with this

only

these

assump-

remind

projection

could

apparatus,

be

the

even

would

opera-

performed, if

corresponding,

apparatus

assume

in quantum mechanics.

every

which

selec-

we shall

of defending

arguments,

assumes

satisfies

of any additional

every

Instead

positive

which

operations,

versa,

particular

what

itself,

are quite usual

always

principle", in

vice

assumptions

describes

into

realizable"

we may state that every by a completely

the absence

sophisticated

one almost

except

B(H) 1

operation.

less

similar

"in

also,

chapter,

mathematically

"physically

describes

instance,

tor E

for

following

properties

reader

mapping

< I for all W s K(H).

criteria

tion

of this

to be described

complex-linear tr(~W)

content

look

nobody

e.g.,

to

like

in

42

§3 The First Representation Theorem

An arbitrary by

effect

an operator

can be represented,

F on the state

space

as shown in the preceding

H

of the system

shall

now derive a corresponding explicit

trary

operation

, in terms

of operators

considered.

representation on H. This

§2, We

of an arbi-

representation

is

provided by

Theorem 1 (First Representation Theorem): For

an

arbitrary

operation

finite or countably

,,

there

exist

operators

Ak,

k e K

(a

infinite index set) on the state space H, satisfy-

ing Z ~ k~K

~

( 1

for all finite subsets K ( K o '

(3.1)

O

such that,

with arbitrary

T E B(H) 1 and X e B(H),

the mappings

, and

** are given by *T

=

I ~ keK

T ~

**X =

~ ~ kcK

X ~

(3.2)

and

respectively.

In particular,

,

(3.3)

the effect F corresponding

to * is given

by F = *'1 =

If the index set K is infinite, the norm

ordering

I ~ kCK

~

"

(3.4)

(3.1) implies that - independently

of K - the infinite sum in (3.2) converges

topology,

while

the

infinite

sums

in

(3.3)

and

in the trace (3.4) converge

ultraweakly,

thus defining the precise meaning of these equations.

Vice

given

versa,

operators defines

an

~

on

any H,

operation

countably

k

~

K,

,, whose

or even uncountably

satisfying adjoint

condition ** and

are given by (3.3) and (3.4), respectively.-

of

infinite

(3.1),

then

corresponding

set

of

(3.2)

effect F

43

Proof:

We

(3.1)

implies

ly many rem

first

k ¢ K;

uncountably

most

countably

bute

in Eqs.

countable

some

of

the

more

that A k can be different

indices

an

discuss

therefore,

infinite

infinite (3.2)

dense

K

to (3.4).

set

if in the second

since

details.

it

may

part

be

operators

of the theo-

replaced

by

f. ¢ H,

an

at

A k = 0 do not contri-

To prove the above statement,

of vectors

First,

from zero for at most countab-

occurs,

subset,

technical

i = 1,2, ....

consider

Keeping

a

i fixed,

l

and choosing an arbitrary natural number n, (3.1) implies

(fi,A~Akfi)

may

be

true

for

finitely

that

= lIAkfiti2 > I/n

many

indices

k

only.

Therefore

the

subset

K. ~ K defined by 1

l;Akfill # 0

is

at most

countably

kJi K i.

By

implies

A~A k < I, and

bounded,

Assume can

(3.5),

infinite.

Akf i = 0

and thus ~

therefore,

thus

natural

be

thus

The

for

same

all

i

then

if

llAkll < 1, for

in the following, after

a

K = {1,2,...}.

set K = {I,...,N}

k ¢ Ki

(3.5)

is

true

k ~ V i K i. all k.

for But

Therefore

the

union

(3.1)

also

all A k are

= 0 for all k ~ U i Ki, by continuity.

identified,

numbers:

if

that K is countably

suitable

ordering,

with

simpler

case

The somewhat

need not be treated

infinite,

separately,

the

and

set

of

of a finite

since we may formal-

ly enlarge K by setting A k = 0 for k > N.

Consider

the operators Fn

Since n,

and

A~A k > 0, satisfy

these F

< 1

d~.

operators for

all

I A~A k • k 0

and

A~(IIXII.1 - X)A k > 0 for all k, the operators ^

Xn

increase

aT.

I A~ X A k k 0 now leads to a sequence of operators

Xn =

~

A~.X Ak.

i< fI; and E I I =

(4.4)

to,

For

is and

the subsystem

because can

thus

one-dimenslonal

the be

states

right

hand

replaced

by,

projection

operators

i fii >< fiI ] , for which

E1 @ Ell = I fl @ fll >< fl @ fi1 I ' (4.2) can then be rewritten, with

=

w - w I @ wii

in the form (4.5)

((fl @ fll )' ~(fl @ fll )) = 0 • This

is true

restriction with

for arbitrary to unit

suitable

(1.13)

scale

for A = 8,

unit

vectors

vectors

is easily

factors.

Applying

f = fI @ fII'

fl e H I and f l l e

removed now

by multiplying

the polarization

The (4.5)

identity

and g = gI @ fII' with gI ~ HI arbi-

trary, we obtain from (4.5)

((fl ~ fll )' ~(gl @ fll )) = 0 • A similar identity,

HII.

66

4((fl ~ fll )' ~(gl @ gll ))

=

((fl ~ (fll+ gll ))' ~(gl @ (fll+ gll )))

-

((fl ~ (fll- gll ))' ~(gl 0 (fll- gll )))

+ i((fl ~ (fll-igll))' ~(gl ~ (fll-igll)))

- i((fl 8 (fll+igll))' ~(gl @ (fll+igll))) ' then leads to ((fl @ fll )' ~(gl @ gll )) = 0 , valid

for

all

combinations

fl' gl g HI

and

fll' gll e HII.

Since

finite

linear

of product vectors are dense in H, this finally implies

A = 0, i.e., Eq. (4.4).

Consider

now,

more

generally,

an effect F I s L(HI)

system I which need not be a decision effect. the particular

case F I = EI,

of the isolated

Assume,

that there exists

as above for

at least one effect

apparatus fI in the equivalence class F I which can also be applied to composite

systems

system

in

II

measures

this

I + II, case.

without

affecting

If applied

in

nor

being

this way,

affected

by

the apparatus

fI

a certain effect ~I e L(H) of the composite system. By this

operational

definition,

the effect ~I must occur in an uncorrelated

state (4.4) with a probability

(4.6)

tr(Fl(W I @ WII)) = tr(FiWl) ,

coinciding with the probability for the effect F I in the state W I of subsystem to

I. Likewise,

effects

FII

of

suitable the

effect apparatuses

isolated

subsystem

II

fII corresponding define

effects

~II e L(H), with

tr(Fll(W I @ Wll)) = tr(FllWll) ,

(4.7)

67

when applied to the composite system. Finally, such apparatuses together.

fl and fll is again expected to measure ~I and ~II

(Thus ~I and ~II are "coexistent"

detail in §6.) The same combined apparatus measure

the combination of two

the effect ~I,II = "~I and ~II"'

if and only if both

effects,

as discussed in

can then also be used to

defined

fl and fll are triggered

single pair I + II. Again the probability

(as above)

to occur

together by the same

for ~I,II'

i.e.,

for the

joint occurrence of ~I and ~II' must factorize in the form

(4.8)

tr(Fl,ll(W I @ WII)) = tr(FlWl).tr(FllWll) for

uncorrelated

states.

Inserting,

in

particular,

pure

states

WI = ~fl >< fl I and WII = I fll >< fi1 I , Eq. (4.8) now implies (4.9)

~I,II = FI ~ FII ' while Eqs. (4.6) and (4.7) imply

~I = FI 8 III '

(4.i0)

~II = ii 8 FII "

(Compare the derivation of (4.4) from (4.2), and note that (4.6) and (4.7) are just particular cases of (4.8).

See also Eq. (3.8) of §3.)

We have thus shown that relations which are already known for decision

effects

form

of

Effects

(e.g.,

Eqs. of

the

(4.1))

(4.9)

and

form

can be

(4.10),

(4.10)

and

generalized

to

arbitrary

(4.9)

will

immediately, effects

be called

FI

in the and

FII.

subsystem

and

correlation effects, respectively, in the following.

For an arbitrary state W e K(H) of the composite system there exists, as is well known, a unique density operator Trll ~ e K(H I) for subsystem I, called the reduction of the state W to that subsystem,

such

that tr((F 1 O

iIl)~)

= tr(Fi.Trll ~)

(4.1i)

68

for all F I e L(HI).

Thus Trll ~ describes the statistics of arbitrary

subsystem I effects in the given state W; or, in other words, describes

the state of the N subsystems

Trii ~

I in an ensemble of N pairs

I + II in the state W.

More

generally,

there

exists

for

any

T e B(H) 1

a

unique

operator

T I = Trii ~ g B(HI)I, which is defined implicitly by requiring

(4.12)

tr(Xi. TrllT) = tr((X I @ Iii)~)

for

all

X I e B(HI).

(4.12)

defines

a

Namely,

linear

for

fixed

functional

T,

the

T(X I)

on

right

hand

side

B(HI) , which

is

of

norm

continuous since

IT(XI) I < ;IXI @ llili IIT;I1

=

IITII1 llXll; ,

and therefore is of the form tr(XiTl) with a unique T I c B(HI) I. Eq. (4.12) thus defines a mapping Trll

: B(H) 1 + B(HI) 1 which, obviously,

is

since,

linear.

It

tr(Xi. Trll ~) > 0 plies

is

also

for

positive

all X I > 0,

TrlI(T*) = (TrlIT)*;

i.e.,

and

by

(4.12),

thus Trll ~ > 0.

T > 0 This

Trll is a real mapping,

implies also

and

im-

(4.12)

with X I = 11 yields

tr(Trll ~) = trT;

i.e., Trll is trace preserving.

Therefore

states

into

(4.11)

Trll

maps

W g K(H)

states

Trll ~ e K(HI) , and

follows as a special case of (4.12). For T = T I @ TII , (4.12)

implies tr(Xi-TrllT) = tr(XiTl).trTll , and thus TrlI(T I @ TII) = trTll. T I

(4.13)

for arbitrary T I e B(HI)I, TII s B(HII) I. This also implies that Trii maps B(H) 1 onto B(HI)I, and K(H) onto K(HI).

A more

explicit

X I = I fi>> 1 times.

Then the

selected subensemble consists of N^ = N tr(FW) = N tr((l @ Fa)WOUt) _

(5.4) ^

microsystems,

according

semble

be

is

effects

to

G e L(H)

to

determined on

these

(5.1). by

Again

the

subsequent

microsystems.

state

W

of

measurements Such

this of

measurements

suben-

arbitrary give

the

result "yes" in ^

N+ = N tr((G ~ Fa)W °ut)

cases,

by

occurence site

defintion

of

the

correlation

of the subsystem effect G @ 1a

systems

selected according

effects

(5.5)

G @ F a . Indeed,

in the subensemble of

to the subsystem

effect

the

compo-

1 @ F a means

85

that both G @ 1a and I ~ F a occur together; and both measurements are performed in the state W out" The probability for the occurrence of the effect G in the selected

subensemble

of microsystems

is thus,

by

(5.4) and (5.5), N+ tr((G @ F )W °ut) w(G) =-~-- = a N tr((l @ Fa)W°ut) Due

to

the

cyclic

interchangeability

of

(5.6)

operators

under

the

trace,

the numerator of the last expression may be rewritten in the form I

1

tr((G @ 1a )(I @ Fa2 )-W °ut (I @ F f ) ) = tr(G.~W) with I

1

~W = Tr ((i @ Fa2)W°Ut(l ~ F 5)) a a 1

1

@ W a )S*(1 @ F a 2)) Tr a ((I @ Fa2)S(W -

(5.7)

'

according to (4.12).

Since the mapping Tr a is trace preserving, 1 I 2-wOUt(l tr(~W) = tr((l @ F a )_ ~ F 5)) a = tr((1 @ F a )W - °ut) coincides

with

the

denominator

of

the

last

(5.8)

expression

in

(5.6),

so

that we may rewrite this equation in the form ^

w(G) = tr(GW)

,

W = ~W/tr(~W)

;

of

microsystems

is

(5.9) ^

i.e.,

the

Moreover,

selected by (5.1),

subensemble

in

the

state

W.

(5.8) and the definition of F, tr(~W) also coinci-

des with the probability

tr(FW) of the measured effect F in the state

86

W,

which

tion

is

the transition

procedure.

(Thus,

probability

as

discussed

N/N

in

for

§2,

the performed

tr(~W) =

selec-

0 again

means

^

that the subensemble

is empty,

-

(5.9).)

be

defined

: W + ~W

via

defined

by

and therefore W need not - and can not

We

(5.7)

therefore

describes

expect

that

a selective

the

mapping

operation

in

the

sense of §2.

To show this, we first

extend

t h i s mapping

~ to B(H) 1 in an obvious

way, by defining 1

i

_ ~r = Tra((l ~ Fa2)S(T @ Wa)S*(I @ F a2))

(5.10)

for arbitrary T s B(H) I. For such T, the operator 1

1

(I @ Fa2)S(T_ @ Wa)S*(l @ Fa2)) belongs Ch.

to B(H) I, since B(H) 1 is a two-sided

1, or

Since

[4]),

Tr a

defines

It depends

maps a

linearly

on T, and

B(H) 1 into B(H) 1 and

positive

tr(~W) < I for

all

linear W,

as

(cf.

is positive

is linear

mapping

shown

ideal in B(H)

and

above.

Therefore

if T is.

positive,

~ : B(H) 1 ÷ B(H) I.

[3],

(5.10)

Moreover,

it only

remains

to

prove complete positivity of ~.

Take,

for

this

Hilbert

space

tensor

product

i.e.,

purpose, H,

and

of

the spaces

an arbitrary consider

Hilbert

the

spaces

(H @ H a ) @ H,

finite-dimensional product is

space

associative

(H @ H) @ Ha,

etc.,

or separable

H @ H @ Ha . and

The

commutative;

can all be identi-

fied with H @ ~H ~ H a in a "natural" and obvious way. Therefore operators like S @ 4, or T @ W a with T s B(H @ ~)I and W a e K(Ha), etc., can be considered as operating on H @ ~H ~ H a . Define now, for arbitrary T E B(H @ ~)I' 1

I

~T = --aTr((I ~ ~I ~ Fa2)(S @ ~)(T @ Wa)(S_ @ !)*(I 0 ~i @ Fa~))

with

Tr

--a

denoting

the

partial

trace

with

respect

to H a

,

(5.ii)

which

maps

87

B(H 0 H 0 Ha) 1 onto B(H ~ ~)i' and which therefore has to be dlstinguished

from the analogous

mapping Tr a : B(H ® Ha) 1 + B(H) I considered

before. Eq. (5.11) is of the same form as (5.10), with Ha, W a and F a unchanged,

but

H 0 H ~ Ha,

1 by

with

H

1 @ 4,

(which is also unitary).

replaced and

by

H 0 H

(and thus

Tr a by Tra) , T by T,

Therefore

H 0 H a by and S by S ~

(5.11) defines a positive

linear

mapping ~ of B(H ~ ~)I into itself. With T ~ B(H) 1 and T g B(H)I, we have

so that (5.11) implies 1

1

~(r 0 T) = Tra([(l 0 Fa2)S(T 0 Wa)S*(I ~ Fa2)] ~ !) 1

i

= (rra[(l ~ Fa2)S(T 0 Wa)S*(l O Fa2)]) ~

=

Here

we

have

~T

used

e T .

(5.10)

(5.12)

and

the

fact

that,

for

arbitrary

e B(H 0 Ha) 1 and T e B(H)I,

(5.13)

Tra( ~ 0 T) = (TraR) ~ T .

(The latter is easily proved, e.g., with the help of suitably generalized versions of Eq. (4.14).)

Taking

H

n-dimensional,

(5.12)

implies

that

~

coincides

with

the

mapping ~n used in §2 to define n-positivity of ~. As ~ is positive for

all n,

this

establishes

complete

positivity

infinite-dimensional H we obtain the (apparently)

of

~, whereas

for

stronger posltivity

property discussed before.

Because Eq. (5.3) results from (5.10) by substituting for F a the unit o p e r a t o r l a , t h e arguments o f t h e l a s t two p a r a g r a p h s a p p l y t o t h e

88

mapping Thus, tors

~

as

well;

indeed,

i.e.,

both

~

is also

linear

~ and $ are operations.

1 @ F aI/2 under

the partial

trace,

and

completely

By rearranging

one obtains

positive. the opera-

two equivalent

but

simpler versions, ~T = Tra((l

@ Fa)S(T @ W a )S*) -

(5.14)

'

or

~T = Tr a (S(T @ W a )S*(I @ F a )) , of Eq.

(5.10).

Such

rearrangements

are

possible

since,

for arbitrary

g B(H)I, X a ~ B(H a) and X ~ B(H),

tr(X.Tr a ((I @ Xa)T)) = tr((X @ la)(1 @ Xa)T) = tr((X @ la)r(l_ @ Xa)) = tr(X-rra(T(l_

@ Xa ))) '

and thus Tra((1

If the apparatus which

have

apparatus simply

@ Xa)T)_ = Tra(T(l_ @ Xa))

f is used

not

triggered

effect

Fa

by replacing

-

to select it

the

-

.

(5.15)

the subensemble

i.e.,

which

corresponding

have

not

operation

F a by F'a = 1 a - F a in (5.14).

of microsystems produced

~'

Then,

is

the

obtained

for arbitrary

T c B(H)I, ~T + ~'T = Tra[(l @ Fa)S(T_ @ Wa)S*_ + (i @ F~)S(T @ Wa)S* ]

=

Tr

a ( S-( T

@ Wa)S*)

= ~T

and thus, in particular,

tr(~W) + tr(~'W) = tr($W) = 1

for

all

W s K(H).

Therefore

~

and

~'

are

complementary

operations

89

(cf.

(2.37)),

and ~ is the non-selective

operation

~ + ~' associated

with ~ and ~', as expected from the construction of the model.

The

adjoint

~*

of

the mapping

~ is defined

implicitly

(cf.

(2.25),

(5.14) and (4.12)) by tr(~*X.T) = tr(X.~T) = tr((X @ ia )(I @ F a )S(T @ W a )S*) = tr((T @ W a )S*(X @ Fa )S) for

arbitrary

with

an

last

trace

X g B(H)

arbitrary

unit

and

vector

in (5.16) with

respectively,

and

T ~ B(H) I. f g H,

suitable

exploiting

(5.16)

After

inserting

evaluating

orthogonal

the

the

T = If >< f I first

and

the

bases in H and H @ Ha,

polarization

identity

(1.13),

we

obtain (f,%*Xg) = ~((f @ g~),(l @ W )S*(X @ Fa)S(g @ g~)) k a ' valid

(5.17)

for all f, g e H and an arbitrary orthogonal basis {gkl in H a .

(Compare

the

analogous

derivation

of

(4.14)

from

(4.12).)

Comparing

this with (4.14), we see that (5.17) can be rewritten formally as

(5.18)

~*X = Tra((l @ Wa)S*(X_ @ Fa)S)_

More

precisely,

when

defined

we have by

proved

(4.14),

(I @ Wa)S*(X @ Fa) ~

(which

(5.18)

with

holds

true

that

can need

this

not

the partical

be

extended

belong

extension.

trace mapping to

the

operators

to B(H ~ Ha)l) , and In

particular,

Tr a ,

the

that

effect

F = #*i corresponding to % is given explicitly by

F = Tra((l @ Wa)S*(I @ Fa)S).

Eqs.

(5.18)

and

(5.19)

are not very useful

in practice,

shall thus not be discussed further.

The results obtained so far are summarized and extended by

(5.19)

however,

and

9O

Theorem 2 (Second Representation Let

Ha

be

Hilbert

a

Theorem):

(finite-dimensional,

space, W a a statistical

Ha, a n d S a u n i t a r y

operator

separable

or

even

operator and F a an effect operator on

o n H @ Ha . T h e n

~T = Tra((l @ Fa)S(T @ Wa)S*)

with

T c B(H) 1 arbitrary,

the o p e r a t i o n

~'

defined

non-separable)

defines

an operation

,

(5.20)

~. With F'a = la - Fa'

by

~'T = Tra((l @ F~)S(T @ Wa)S*)

is

complementary

to

~,

and

the

non-selective

(5.21)

operation

~ = ~ + ~'

associated with ¢ and ¢' is given by

~T = Tra(S(T @ Wa)S*)

Vice versa,

given any two complementary

there

a Hilbert

a

exist

unitary

operator

represented also

by Eqs.

require,

(i.e.,

a

S

in

space Ha,

(5.20),

addition,

one-dimensional

operations

operators

on H @ Ha,

such

(5.21) that

projection

~, ~' and

(5.22),

is

~ and ~' on B(H)I,

W a c K(Ha) , F

that

and Ha

(5.22)

.

e L(Ha) , and

~ = ~ + ~' are

respectively.

separable,

operator),

Wa

and

a Fa

One may

pure a

state

decision

effect (i.e., a projection operator).-

Proof: that

The

first

part

of

the theorem has already

been proved.

(Note

the dimension

of H a in the above model was arbitrary.) To prove the second - and more interesting - part, we start from the representations (3.19) and (3.20) of ~ and ~', ~T =

as provided operators

~ Ak T ~ keK

by Theorem

Wa,

,

~'T =

I', and construct

Fa a n d S e x p l i c i t l y .

For

~ Ak T ~ keK'

,

(5.23)

the Hilbert space H a and the the

sake

of

definiteness,

the

91

index

set

J,

of

complementary without

which

subsets,

loss

of

the

is taken

generality

by

nal index set J was finite,

Take

for

Ha

a

sets

and

K'

occuring

to be J = {1,2 .... }. This setting

(5.13)

are

can be done

Ak = 0 for k > N if the

Hilbert

space

with

origi-

orthogonal

Then H ~ H a = H_ may be decomposed, =

in

J = {I...N}.

separable

{gia I i = 0,I,2...}.

K

basis

in the form

@ (H 8 g~) = @ H i , i)O i~O a

orthogonal

into

identified

subspaces

with,

H.

Hi = H ~ g i

Taking

the

isomorphic

subspaee

H

to,

~ H

and

apart,

therefore

we

may

also

O

write ^

H _

=

H

•

^

H

,

H

=

e

Hk

,

Hk

~ H

•

k)l In

the

following,

values

the

i = 0,1,2...

indices

i and k are

and k = 1,2...,

always

which

somewhat

assumed

to take

simplifies

the

the no-

tation.

Define,

in terms

of

the

operators

Ak,

k = 1,2...

entering

(5.23),

an

^

operator

A : H ÷ H = ekH k by Af = @ ~ f K k

(Note

that,

for

arbitrary

f e H,

.

Akf e H ~ Hk.)

Eq.

(3.18)

of

Theo-

rem I' then implies

(Af,Ag)

i.e.,

A

is

= ~ (Akf,Akg) k

isometric,

with

= ~ (f,~Akg) k

operator

norm

= (f,g)

IIAII = I.

;

Its

adjoint

^

A*

: H ÷ H, defined

by

(f,A*f) ^

= (Af,f)

^

for all f c H and f e H, is also bounded,

with

(5.24)

92

i(f,A*)J IIA*II = Sup

^ Ilfll

0(Af,f) = IIAII = I .

= Sup

Ilfll

Ilfll

II II

Isometry of A implies ^

A*A = 1 ,

AA* = E

^

with

some ^

projection

operator

(5.25)

^

E

on

H.

Indeed,

A*A : H ÷ H

and

^

_AA* : H ÷ H are obviously self-adjoint,

and A*A = 1 since

(f,_A*Ag) = (Af,Ag) = (f,g)

;

thus ~2

Explicitly,

^ = AA*AA* = AA* = E .

A* is given by

A*

: • fk k

÷

(5.26)

~ ~fk k

^

for arbitrary @kfk e H (i.e., fk ~ H, ~k;Ifkli2 < ®), since then

(A*(O fk),f) = (~ ~ f k , f ) k k

= ~ (fk,Akf) k

= (0 fk' • Akf) = (0 fk' ~f) k k k i.e.,

;

(5.26) is satisfied. ^

Vectors ! e H = H • H may be represented

^

with

f e H and

in matrix notation as

^

f e H.

In this notation,

be written as operator matrices,

=

Xl

X2)

X3

X4

bounded

operators

X on H can

93

with bounded operators ^

XI

: H

÷

H ,

so that, according

X2

^

: H

÷

H ,

X3 : H

+

^

H ,

X4 : H

^

+

H ,

to "matrix multiplication",

(xif+x2 ) x f= ^

X3f + X4f The adjoint of X is easily seen to be

x(X while

operator

products

to

be

calculated

ordering

of

the

tion",

preserving

ments",

of the operator matrices.

We

define,

now

in

the

are

this

matrix

S = -

Obviously S* = S; moreover,

notation,

A

the operator

-S

multiplica"matrix

ele-

on -H - - H

@ Ha

. ~

-

(5.27)

AA*

S is also unitary:

s.s=ss,=s2= ( . . . . .

"matrix

(non-commuting)

(0

by

by

A*(~-~*)

A*A (I

-

AA*)A

./j~* +

(i" -

)

_A/j*) 2

(i 0) 0 Here ^

we

have ^

used

the

~

=i_.

following

facts:

By

(5.25),

^

1 - AA* = 1 - E is a projection operator,

A*(I-~*)

=A*-A*~*

so that

= A * - A * =0 ,

( I - ~*)A= A-~*_A= A-_A= o ,

A*A = i,

and

94

and AA* +

- .AA*) 2 . = .AA*. + . i^ - A A * .

(1

--1

= I^ .

a

For W a we take the projection operator ont 9 go' a

a

Wa -- I g ° >< go I .

Then we obtain,

(5.28)

for arbitrary T e B(H)I and f e H,

(5.29)

(T 0 Wa)(f O g~.) = ~jo(Tf @ goa) .

a

Due

to

the

may

also

isomorphism

be

written

H @ H a -- eiHi,

as @i~ji f.

In

H i -- H,

the

the vector

above

matrix

fj = f 8 gj

notation,

this

means f = -o

, 0

f. = -J

for j > 0 , jk f

so that (5.29) takes the form

(T @ Wa)f ° =

(Tf)

(T @ Wa)f j --

(0)

Since

vectors

of the

for j > 0 .

0

0

form -j f. span _H, this means

that

T @ W a has

the

matrix representation

T wa (T0 0°I Define

now,

for

an

arbitrary

(finite

or

(5.30)

infinite)

subset

^

J = {1,2...},

a projection operator EK on H by ifkeK E (e fk ) = • gk ' k k

gk = $ fk L 0

ifk~
< gka I , kgK

Ea O,K

=

I goa >< goa I

+

E aK

K

of

95

on H a . Then,

by arguments

similar to the ones leading to (5.30), one

easily verifies the relations

(0

1 @ Ea

--

,

0 A

Ea

straightforward

1 ~

(i o)

=

O,K

0

K

E

(5.31)

"

K

calculation

with

(5.27),

(5.30)

and

(5.31)

then

yields dE. (i @ E~)S(T @ Wa)S(I @ E~)

a

=

~-

(i @ Ea,K ) s ( T-o

@ Wa)S(I- @ Eo,K)

(0 01 ^

0

(oo)

^

K--

--

(5.32)

^

E ATA*E

u~.

0

B

K

Since T e B(H)I , we have B e B(H) 1 , and thus TraB e B(H) 1 ial

trace

with

the

can be calculated particular

basis

by

(a suitable

{g ~}

in

Ha

analogue

used

The part-

of) Eq.

before;

for

(4.14)

arbitrary

f, g E H we then get

(f, TraB g) =

~ ((f @ g~), B(g ~ g~)) .

(5.33)

j~o As above, we have in matrix notation

(g) 0

(0) ~k 6 jkg

and similarly with f instead of g. Thus (5.32) and (5.33) lead to (f, TraB g) =

= By definition,

E

j ~ ~. Therefore

~ ((O 6jkf), B(O 6jkg)) j>0 k k ~ (A*E (~ TA*E (O ~jkg)) j>O - ~ k ~Jkf)' - K k

reproduces the operators

•

@ ~ j k f if j g K, and annihilates E

it if

may be dropped in the last expres-

96

sion,

if the

sum

over

j is restricted

to the

index

set K. Eq.

(5.26)

implies A*(@ ~jk f) = A@f -

so that,

k

J

finally, (f, TraB g ) =

~ (A~f,TA~g)=

j~
< gk i + I go >< go I + ~ I gk >< gkl ks K kcK ' a a = ~ I g i >< g i I = 1 a . i>0

Since

97

As announced projection

in

the

theorem,

operators,

and

the

Wa

as

operators defined

Fa

by

constructed

(5.28)

here

describes

are

a

pure

could

also

state.

We

finally

remark

be carried if

the

that

the

above

contruction,

through with a Hilbert

set

of

operators

Ak,

obviously,

space H a of finite dimension

k E J

entering

(5.23)

is

N + 1,

finite,

i.e.,

J = {I...N}.

The

theorem

just

proved

allows

some

conclusions,

which

shall

be dis-

cussed now.

We have

shown,

apparatus

and

measurement tions

to

its

of

~ and

ding

first,

the

effect

second

~ and

a suitable

apparatus

choice

of

Fa

the

of

This, red

part

~'

does

and

not

could

complementary

opera-

Conversely,

accor-

the

two

measurement.

theorem,

initial

prove

be

any

at least

already

interacts

formally,

that

such an apparatus

reproduces

possible

the very

S

general

this,

2 provides

F e L(H) insists by

not

only

describe in

possible

"conventional"

postulating

projection

projection

that

operator,

the the

operators

yes-no quantum

"pointer" effect

E,

support but

for

rather Indeed,

mechanics

the

effect

F a is always

F measured

by

such

our

all

measurements. for

In

model

class

Besides

that

the requi-

concrete

in a much more abstract way in §2.

tion

H 8 Ha .

it is, neverthe-

as introduced

additional

effect

laboratory.

rations,

Theorem

on

-

with

the


the simplest

with the

"pointer"

operator

in

by

by an appropriate

W a and

constructed

the

"in principle",

which

state

describes

of complementary

scattering

actually

that

pair

the

consistency

satisfactory

measurements

and

way - i.e.,

of

of an effect

indeed

"realized",

Ha,

model

microsystems

this

the

be

space

view of the internal

yes-no

of

mechanical

of the type considered,

state

interaction

quite

with

could

apparatus,

clearly,

less,

with

in an appropriate the

quantum

F e L(H)

associated

operations

microsystem

our

interaction

an

~'

that

of ope-

assump-

operators

even

model

of

if

one

apparatus

described

by a

an apparatus

need

98

not

be a projection

F E L(H)

can also

operator.

be measured

apparatus,

according

operations

~ performable

ly

general,

e.g., not

the be

so

F

condition

and

whether

or

states

not

is

and, (2.9)

-

e.g.,

cular

not

that

time

(2.8)

by

During

system,

and

formalism,

this thus

quoted

evolution

which

is

picture

applies

again

- leads

of a

above the

closed

system,

We have

shown,

very

measurements

One

and

argues,

for

described,

in the SchrS-

pure

states

-

(2.8)

(as

states,

into

whe-

mixtures.

Not only the parti-

pair of complementary

from

the

certain

of

the

interaction interval

choice

the "reading" the

(isolated)

system and

is

indeed, apparatus described

reduction

moreover,

as

of

the

time.

of the appara-

microsystem

systems

of

- or even the

can not be described

to a state W °ut whose

the state ~W.

to

into

every

a

is misleading;

composite

look

the non-selectl-

interval

to closed

of

In these models,

describing

time evolution

ef-

quantum mechani-

is independent

interaction

of

by a suitable

during

Fa

not

mechanics

pure

in fact

apparatus

effect

its

which

t h e argument

just

the

can

apparatus

formulae

is unfounded.

results

~

would

mechanics.

most

process.

kind)

initial

due

states

simply

(5.22),

- of

but

(as,

independent

for the state vectors pure

~

classes

are

reduction"

can be reproduced

the

Wa

completeon

first

the

as

changes

quantum

reasoning

(2.9),

~ = ~ + ~'

(Note

tus.)

and

in

the

the

nature of the apparatus.)

quantum

transforms

of the measuring

with

by

transforms

this

~ and ~'

that,

states

packet

equation

(2.9)

that

microsystem

existence

"wave

of

the model

pure

operator

Moreover,

restrictions

admitted

state

evolution

always

2 shows

operation

time

be

operation

cal model

by

2.

are also

(Likewise,

are

to

that

explained

operations

ve

claimed

can

cases

either.

states

the

the

apparatuses

in view of the macroscopic

picture)

Theorem

way

pure

Theorem

additional

~ described

by the SchrSdinger

dinger reas

this

particular,

instance,

of

every effect

by such a "conventional"

for measurements

restriction

frequently in

part

by such model

~*F = F

in

a

last

conceivable

only

W a . But

"in principle"

the

operations

natural anyway,

It

to

that

justified

fects

On the contrary,

is a n

open

by the usual

only.

Therefore

shown

here,

plus here

microsystem in

the

-

Dirac

to the microsystem

that the selective

the

is

opera-

99

tions

# and #' can be also understood

in terms of conventional

quan-

^

tum mechanics.

Since

produced

are

by

effect Fa, composite a

~

always

in,

preceded

e.g.,

by

the

the

state

reading

W = #W/tr(#W)

of

the

apparatus

one is in fact dealing with correlation measurements system.

convenient

ments.

measurements

This

In our model,

"shorthand" description

apparatus variables mapping

Tr a . We

belief,

state

description

is obtained,

of as

~ and #'

such in the

just provide

correlation

measure-

case

when

of ~,

the

are eliminated with the help of the partial trace

may

thus

changes

well understood,

the operations

on a

of

conclude the

that,

type

(2.8)

if quantum mechanics

contrary and

to

(2.9)

a wide-spread

can be perfectly

is assumed to be valid also for

measuring instruments.

Our

model

complete

also

positivity.

completely

positive,

scribed by Eq. for

illustrates

all

in a simple way

An operation if

(5.10) and

(5.11) are positive

(5.11), an

~ of B(H @ ~)I

for arbitrary

Hilbert

spaces

one immediately realizes

operation

preted

as

acting

on a composite microsystem which

ing subsystems

performed

physical

~ as given by Eq.

the mappings

finite-dimensional)

the

by

meaning

(5.10) into

above

itself

is de-

(and, in particular,

H.

Comparing

now

Eqs.

that ~ may also be inter-

the model

apparatus

consists

f,

but

now

of two noninteract-

I and II with state spaces H and H, respectively,

is thus described

of

in the state space H @ H. The model apparatus

and f is

the same in both cases,

since the state space H a , the initial apparatus state W a and the "pointer" effect F a are identical in Eqs. (5.10) and (5.11). Moreover, the unitary operator in (5.11) describing the "scattering"

between

the composite microsystem

I + II and the appara-

tus f is given by S @ ~. Obviously this means that, first, the apparatus

interacts

fected,

and second,

apparatus since

only with subsystem I while

the

is

independent

resulting

same operator

that the interaction of whether

"scattering"

is

leaving between

or not

S on H @ H a . This interpretation

of our quantum mechanical

model

is therefore

subsystem

subsystem

described

--

subsystem II unaf-

in

II is present,

both

of Eq.

I and the

cases

by

the

(5.11) in terms

in full accordance with

the physical meaning of the mappings ~, as discussed in detail in §2.

100

In spite

of its considerable

overestimate reasons

the

to

doubt

appropriate

This is a highly

nor

require

really

latter

should

usually

easily

though into

called

quantum

attempts

mechanics,

have have

the

been

resulting

our

model

would

interest. trons. an

be

If

now

space

totally

trons.

still

be

For instance,

electron,

state

satisfactory

mechanics

In

measuring

then for

the

combined

antisymmetric view

of

process

contains

all

should

product system,

with

are

validity the

also

of quantum

system

described

in

rather than a macroscopic

to

quantum

description

of Theorem

true

are relatiAl-

such

ideas

yet be consi-

macroscopic

cases

of

sy-

practical

contains

too,

or

elec-

is itself

of

the

model

as a model only,

elecof

the

rather

of actual measurements.

2 which

space f.

set of

for macrosystems,

mechanical

for macrosystems.

apparatus

of

permutations

be considered

state

in

The

properties.

can not

electrons,

really

the

system.

which

instrument

our

mechanics

majori-

observed

incorporate

in many

this,

applications

subsystems.

H @ H a is not the appropriate since all state vectors have to

respect

than as a complete and realistic

There

to

literally

oversimplified

tensor

the

"classical"

every real measuring

the

in-

of the universe),

of

description

were

the microsystem

measuring

atomic

ones,

theories

in spite of some partial successes.

quantum

the

body as a compo-

the size

made

stems,

if

fully

is

in terms of a much smaller

suitable

good

since their measure-

behaviour

to

even

a

at all,

dered

But

yield

1024

the "macroscopic"

and

e.g.,

are

form

system are neither

actual

be describable

There

present

like,

exceeding

the

one should not

since the overwhelming

a complex

for

its

about

not even observable

measurable

numerous

least)

"instruments"

rather

in

a macroscopic

description,

such

significant

observables, vely

of

however,

the model.

systems

describes (at

redundant

(and perhaps

ment might

of

of

mechanics

o f macroscopic

"observables"

practice

quantum

consisting

value,

significance

Quantum mechanics

system

ty of

that

theory

struments. site

physical

heuristic

Ha

do not presuppose

the

In such applications, is

also

a microsystem,

i01

Denote

by s and a the microsystems

tively,

and

assume

tering

system

that,

when

put

together,

they form a binary

scat-

s + a. If N pairs s + a are prepared in an uncorrelated state W @ W a = W _ in , their state after the scattering is

incoming

wOUt = S win S,

with

above,

the N systems

and thus

Keeping

the

initial

procedure

yields

system

as

s,

systems

of

state

operator

Eq.

of

into

two

varying

2.

measurement

F a performed then

"yes" at

W,

as

this

for

Moreover,

complementary

to the outcomes

One

H @ Ha,

~ : W ÷ W

Theorem

respect

scattering.

on

in the state W = Tr aW °ut.

operation

(5.22)

separated

the

S

of system a fixed while

nonselective

by

with

after

scattering

s are finally

Wa

a

be

a yes-no

s + a

unitary

described

s can also

tively,

a

again

they are selected

pair

with state spaces H and Ha, respec-

the

the N

ensembles,

or "no",

subsystem

concludes

if

respec-

a of each

as

above

that

W = ~W/tr(~W)

and

^

these

two

subensembles

W' = ~'W/tr(~'W), Theorem

2,

and

with

are

~

consist

and of

in ~'

the

given

states by Eqs.

N+ = tr(~W).N

and

(5.20)

and

(5.21)

N_ = tr(#'W).N

of

systems,

respectively.

An

"indirect"

measurement

complementary ding

effect

strument by Wa, but

F = ~*i

w a for systems

insensitive

system

apply

composite perform

or

operations

F will

regardless

restriction

are to

s. The

combination

also

may

be

of

a

fa

the

~' not

or not

projection and may

the

and

of the corresponof a preparing

together

in-

be

to

and

a,

Then,

effect

F,

~ = ~ + ~'

be described

as a single

system s in the follow-

let it

experiment.

measures ~,

to a single

apparatus

two

the effect F a on system a

considered

Wa,

yields

a single system a prepared

fa measuring

from

not uncommon

is unnatural,

of the theory.

s,

the

in general

of whether

measurements

ments

f

therefore,

a in the state Wa,

system

"no"

apparatus the

system

f, to be applied

then

type,

~ and ~' and a measurement

system

the

"yes"

this

apparatus

release

s,

outcome

a

to

apparatus

way:

effect

on

and an effect

effect ing

operations

of

at

as

scattered read shown

and

may

system

on

at

the

fa

the

above, be s.

by a projection

used Again

this to the

operator,

this

is true for F a . Since such indirect in practice, this shows once more that

operators in fact

as

lead

describing to internal

yes-no

measure-

inconsistencies

102

It has even been argued

([13], Ch.

II) that typical measuring instru-

ments always contain a suitably prepared microsystem a (a "trigger"), which

first

eventually

interacts

triggers

with

could

describe

observed

some observable

pic part of the apparatus we

the

the

s,

and

afterwards

change on the remaining macrosco-

(the "amplifier").

interaction

system

between

Taking this for granted, a and

s as

a

scattering

process, uring

and consider the "amplifier" as an effect apparatus f a measa certain effect F a on the "trigger" a, thereby arriving at

exactly the situation considered above. Whether or not such a description of the quantum mechanical measuring process listic

and

really

helpful

discussed here, however.

for

a

deeper

is sufficiently

understanding,

rea-

shall not

be

103

§6 Coexistent

A

set

of

effects,

coexistent applying

are

either

ratus

apparatus here

"yes"

or

in

tr(FW)N

combination ment

one

"no",

and

of effect

of all effects

The

each

which

-

effect respond

to the effect

apparatus

F E C when

c may

f, which

with

set

of

together

by

as having

whose

the

i.e.,

outputs

appropriate if the appa-

in a state W,

F e C gives thus

be

perform

the combined

a

Such an appara-

F e C,

to N >> 1 systems

apparatuses

or:

be visualized

of microsystems;

corresponding cases.

c - may

for

successively

channel

coexistent

F e C can be measured

label

to ensembles

c is applied

"yes"

called

to single microsystems.

by the

channels",

frequencies

the output

is

- if all effects

a suitable

"output

relative

C = L(H),

effects

tus - abbreviated several

Effects and Observables

then

the output

considered

as a

the joint measure-

apparatus

c is applied

to

a single microsystem.

In

conventional

operators)

quantum

are

are usually a

considered,

with

out

to be equivalent

ries, rable. trary

slightly

features

that

there

We want

only

decision

coexistent (Ludwig

meaning;

of

quantum

are sets

theory,

now

sets

of

(projection

decision

[2] defines

effects

"commensurabili-

however,

this

It is one of the most as

of decision

effects

finally,

to "coexistence".)

The notion

compared

to

classical

turns charactheo-

effects which are not commensu-

the corresponding

of "coexistence"

problem

as defined

above

for arbiis due to

[2J.

Commensurability verbis,

narrower

to investigate

effects.

Ludwig

and

called commensurable.

ty"

teristic

mechanics

or

at

in quantum least

tentatively

measurability.

Nothing

definition

coexistence;

of

stem with all parts of the different

mechanics

of this

-

sort

neither

as

channels

considered

synonymous

is implied, the

of the "composite"

output

is often

need

with

apparatus be

"simultaneous"

however,

interactions

- expressis

of

by the above the microsy-

c, nor the responses

"simultaneous"

in any sense.

104

This

may

be

apparatuses effects

F

(Note

that

the

interacts whole

apparatus

simple

f acts

is

If

and

earlier

occupy

can

put together,

and applied

single

microsystems.

The

single

apparatus

channels",

this

c.

Ig

f and

do not be

g,

apparatus

c

Ig.)

and

that

the

with

the

in the speciis relevant

the microsystem we assume

Assume

also

that

that

the

the apparatuses

f and g

Then both apparatuses

f and g

successively

combination

which

respectively;

than

effect

with the appara-

are included

during

region in space.

Having,

measurement

f and g. What actually

non-destructively,

the same

the

two

would measure

than the measurement

apparatuses

If

that

of application

the apparatuses

Consider

separately,

Assume

earlier

intervals

interval

example.

when applied

the times

time

with

a

respectively.

of the effect

are

the

G,

f is performed

fication here

by

f and g which, and

apparatus tus g.

illustrated

f + g

- first may

by

construction,

is

expected

to

be

two

f, than g - to considered

different

measure

two

as

a

"output

coexistent

effects F 1 and F 2.

These te -

an

the

F 1 and F 2 are easily determined.

probabilities

i.e.,

for

of the parts

arbitrary

stems f,

effects

in this

triggering

nonselective state

state

the

W.

When these

it

tr(FW)N

in

operation

c

is

applied

systems

first

cases.

performed

therefore

After

by

f,

cases.

lities

and

question

the

are

tr(FW)

two "output

successively interact this,

to

with

with

the

Since,

channels"

c - by microsystems

the N systems

triggering

tr(GW)N = tr(G.~W)N = tr(~*G.W)N in

of

f and g of the apparatus

state,

W = ~W,

triggering

To do this, we calcula-

~

N >> 1 sy-

the apparatus denoting

are

tr(~*G.W),

the

the

in the

apparatus

therefore,

in

g

new in

the probabi-

effects

measured

together by c are

F1 = F ,

According obviously,

to our

definition,

F 2 = $*G •

these

they are not measured

two effects

(6.1)

are

"simultaneously".

coexistent

- but,

105

If,

in particular,

both F and G are d e c i s i o n

F =E

and the a p p a r a t u s

f performs

,

G=E

an "ideal"

effects,

,

measurement

(cf.

(2.9)),

~W = EWE + E'WE'

w i t h E' = i - E, then

(6.1)

reads

F1 = E ,

In

this

only

if

[E',E]

case E

F2

is

also

a

and

E

commute.

F 2 = EEE + E'EE'

decision

effect

Namely,

if

•

(6.2)

(and

equal

[E,E]

= 0,

E)

we

if

also

and have

= 0, and thus

F 2 = E2E + E'2E = (m + E')E = E

Vice

to

versa,

if

H = F2E = EEE commuting

F2

is

since,

a

by

projection

projection (6.2),

operators

operator,

F 2 commutes is

again

the

with a

.

E,

same and

projection

is

true

the p r o d u c t operator.

for of The

operator A = EE - EEE then s a t i s f i e s A * A = EEE - E E E E E ~

which

finally

= H - H2 = 0 ,

implies 0 = A = A* = A* - A = [E,E]

As

a

"physical"

volumes ly, stic

V 1 and

therefore functions

example, V 2 and measuring

consider

operating -

when

two at

counters,

times

applied

t I and

.

occupying t 2 > tl,

separately

-

the

spatial

respective-

the c h a r a c t e r i -

106

E = XvIX(tl))

of

the

position

operators When

at

applied

operators

different

times,

after

first

ures

an

effect

that

the

counters

not

be

X(t)

the

F2

which

satisfied

by

of

counters

involved,

and

most

at

t = tI

thus

-

t = t 2.

But

position

do not

commute.

one,

therefore,

the second

not

a

effect.

decision

counters,

if

likely

and

also E and E,

instantaneous

actual

scription

E = X.v2(X(to))

and

is

perform

,

and

possible

would

at

show

A

all

that

(Our

"ideal"

however.

assumption

measurements more

-

even

counter meas-

will

realistic

would

be

de-

much

more

counter

does

commute,

they

a single

not measure a decision effect.)

On

the

can

other

hand,

be measured

of

the

type

decision

if two projection

together-

Indeed,

and

sufficient

for

quantum

mechanics.

A

ry

at least

considered,

effects.

operators

and

"in principle"

therefore

should

commutativity the

and

known

of

rigorous

- by an apparatus

describe

is well

commensurability

complete

E and E

commensurable

to be neccessa-

decision

proof

of

effects

this

in

criterion

will be p r e s e n t e d below.

In order

to obtain

the

case

of

if

there

two

considered brevity,

By

to

output versa;

The

two as

F I and

apparatus

outputs

results

some yield

-

c

both

F 2.

criterion,

They

are

yielding, either

of measurements

wiring

an output

if

of

electronics, yes-no

which

is

2

and

only

are if

"yes"; at

realization

and

least of

an

one

or

"no" F 2.

besides output is

output the

if and to

a

- which For

the

only

single can

be

sake

of

i and 2 here.

the apparatus

if

of

applied

F 1 and

outputs

"yes"

we first consider

coexistent

when

"yes"

1 I% 2 = "i and 2" which

I and

technical

and

additional

I' = "not I"

outputs "yes"

an

coexistence

these two outputs are simply called

adding

fied

effects,

exists

microsystem,

a general

1

"yes"

c can be modi-

1 and is

2;

"no",

e.g., and

an vice

if and only if both

1 V 2 = "I or 2" which outputs

the corresponding

i and

new output

2 is

is

"yes".

channels

in

107

terms

of

well

the

known

to

continued, more

already

existing

every

thereby

complicated

channels

experimentalist. leading

when

also

expressed

for This

to

new

in

terms

the

outputs

procedure

outputs of

can

which

the

1

and

be

look

2

is

further somewhat

original

outputs

1

and 2.

As

is

also

operations

well

known

with

outputs

"or"

satisfy

rules

of a Boolean

usual

symbols

y,

etc.

(and

the

denoting

from

characterized

rules

of

by

ordinary

algebra.

' for

obvious

arbitrary

for

the

and

the

for

the

"and"

and

calculational

already

v

the most

examples),

"not",

i.e.,

we have

"and"

outputs,

given

words

logic,

(Therefore

"not" , A

the

introduced

"or".)

important

With

the

a,

B,

calculational

rules are : ~ A ~ = ~ ,

~ A B = B A ~ ,

({l A 8 ) A y = {~ A (IB A y )

and analogous

rules with

A

v

for

; (6.3)

(=')'

= =

,

(=

~ A ~' = ~ ,

vB)'

=

~ V~'

~' ^

B'

= I ,

,

(= AB)'

=

~' V

S'

,

~ A I = ~ V ~ = ~ ;

and

-~

(the puts:

distributive I,

which

Corresponding

^.(BvY)

= (~A

B) v (~ ^ ~ )

v(BAy)

= (~v

B) A ( ~ V ~ )

laws). is

In

(6.3)

always

output

there

"yes",

channels

and

are

occur

"l.

/ the

~ = I',

also

,

two

which

easily

(6.4)

"trivial"

is

always

incorporated

out"no".

in

the

given apparatus.

Starting

from

the original

well-known

technical

thus

lead

to

This

"enlargement"

to a "natural" in each single

manipulations

modified

end,

apparatus

of

apparatuses the

apparatus,

however,

experiment,

leading

yields

c with as

only

two output

symbolized

with

additional

when finally

pursued

by

' , A output

far

and

v

channels.

enough,

to an apparatus

the 16 outputs

channels,

comes

b which,

108

I,

1 A 2,

2,

I' ,

1 A 2' ,

(i A

I v 2',

using

that

the

any

further

16 outputs ment

calculational

of

does not enlarge

the

duplicate

apparatus

already

realized (I A

1 A

((I A

Being

V

, the

As

all

set

1 and

the outputs

16

with

~ c B are 2, B is

in fact

look

different

ting

the

channels from

other

when

I

Therefore

we treat

here

cases

it may

these

outputs

every

single

F 2 = 1 - F = F'

coincide

experiment. are

and

(i.e., For

coexistent

to these

so that a further

enlarge-

new

outputs,

2

to

the

but merely

either

all

every

(e.g.,

"yes"

the

F E L(H):

algebra.

or

effects

to the

by them.

technically

themselves

and

containing

(6.5)

nevertheless,

instance, for

algebra

outputs

suitable

happen,

' , A

operations

generated

realized with

I = 1',

is a Boolean

Boolean

the

2))

2',

these

algebra

to

(1' A

operations

to (6.9)

the outputs

are

(I A

= I'A

by applying

the Boolean

outputs

In particular

checks

v

2')) V

the smallest

original

rent.

easily

' , 2k and

(2 V 2 ' )

(6.5)

corresponding

each

one

2') V ~ = 1 A

respect

obtained

I and 2 - i.e.,

output

devices.

closed

(6.8)

(6.4)

yield

(I A

2') = I' A

B of the 16 outputs

outputs

outputs

The

thus

(6.7)

2',

= (I A

etc.)

2),

(For instance,

2)' = I' V

2) V ( I ' A

and

list,

2') V (I' /k 2)) = (I A

(I' A

A

(6.9)

really

ones.

(6.6)

I' V 2 ' ,

operations

the given not

2,

(6.3)

of the

would

2 ',

~.

rules

application

1'A

2') v ( l '

I' V

I,

By

(6.5)

1' 2k 2,

(I A 2) v (l'/k 2'),

I v 2,

2' ,

by

to

(6.9)

connec-

electronic)

also as diffethat all

some

of

"no")

in

F1 = F

If an effect

and appa-

109

ratus of

f with

the

second

apparatus

output

channel

an

above,

It is obvious, always

which

less

than

dered,

16

the

output

also

treat the

~

and

single

experiment,

(E.g.,

the

Such

four

a v 8 when

shall, = (~V

advantage

of

want

however,

an

not

(~' v

output,

More

generally,

al...

an

to

use

with

(6.9)

an

is

~i''" an.

generalization

b with

just to

as

consi-

(6.9)

the

two

apparatus

(a 2 V

...V

"universal"

said

never to

each

(~2v

Boolean

and

narrower

meaning

a ~ 8

8)

the

form

in

aI v

each

other,

the

disjoint

n = 2

to

each

shall

be

of

for

with

above

an) = (a I A

a2) V

also

union

n > 2

... V

... V an ) = al ~" (a 2 V

(a 1 A

... v

In

of

is

the

natural

an) = ~ ,

a n)

, etc.

this

written (We

~ or 8"

~A

under

a

other.

8 ~ ~. (6.7).)

a 2 V ... V a n, is

in

"or".

"either

cases

listed

"yes"

other.)

a or 8",

is

from

exclude

exclude

8 = 2,

called

both

and thus aI V

Our

different

by (6.4), aI A

to

"tri-

c.)

formally

same

are

notation

of

an,

This

are

excluding and

B--~

8') V (~'/~

a I ~" a 2 ~> ... ~

the

alge-

cases.

this

output

pairwise

the

only

b

identify

"output"

(6.5)

that

to

this

a = 1 and

so

apparatus to

example

as

channels.

apparatus

list

(6.5)

"either

8') = ( a A

the

mutually

stress

an

the

to

a A

(6.6) means

(In

~,

tempted

by

and

enlarged,

output

Boolean

added

therefore

outputs

actually we

8) A

and

at smaller

yielding

8 satisfying

be

be

B also in such exceptional

outputs

~ 8

outputs

F 1 = F,

I' and 2 of this

reduces

to

f, so that f

for

16 different

might

I and

corresponding

then be further

channels.

I',

A

1 = a

technically

I' = 2

have

all

one

realized

i,

outputs,

the outputs

output

outputs,

however,

case,

be

F'.

a' = "not a"

to the apparatus

b with

arriving

the output

measure

c may

cases,

thereby

channels

to

algebra

such

identification

vial"

decision

In

different

different

two

apparatus

that

to

be added

yielding

however,

may

four

Two

c

F, then

used

thus

This

outputs,

bra,

may

be

to an apparatus

coincide.

coinciding

a measures

may

apparatus

for F 2 = F'.

described

has,

output

same

becomes 2 = a'

yes-no

with

written effects since,

110

When

an

single red

apparatus

b

mlcrosystem,

as

the

result

of

the

type

each

one

of

of

a

measuring

a certain

apparatus

b thus measures

form F

a certain

subset

@ FS, B contains

at most

above

-

correspond

puts

do

effects

B of effects nal

in

may nevertheless

outputs

i and

effects 2,

and

tus

realize

said

to

joint

measurement

to

the

result

a

of

but possibly

in the example But

even

experiment,

The

F , ~ e B, which

not necessarily

effect.

the

imply fewer.

discussed

if two outcorresponding

as we shall see later on.) The set and it contains

corresponding

Therefore

coexistent

is

same

F2

pletion of the original b

- as

single

respectively.

effects

effects,

by definition,

F1

i.e.,

As ~ @ 8 does

be equal,

applied

~ e B can be conside-

-

these

outputs

every

is coexistent,

coexistent

all

to the

been

- on the given mlcrosystem.

16 different

coinciding

coincide

16 outputs

e L(H)

B of L(H).

two

not

F

has

measurement

together

(In particular, clearly

its

yes-no

effect

considered

to

the

particular

we call B a coexistent

set of effects this

the origi-

the appara-

{FI,F2};

coexistent

com-

completion

B

of

{FI,F2}.

Although

the

visualized an b

- and therefore

"enlarged" is

not

and

~

really

are

vial"

apparatus

outputs

calculated (For

from

the

instance,

if

microsystem

yields

outputs

2

IA

(i A 2) v (I' A nal F

apparatus

the actual

and 2') c

for

outputs

i' A

for

2'

in

in (6.7)

already

e B

purpose.

under

I and 2 of of

this

output

1

(6.6)

are

is also

measures,

and

"no", at

The

of

this

easily

apparatus

"trivial" and

(6.5)

all

I

"nontribe

the original

apparatus

c.

apparatus

to

"no"

etc.)

to

outputs

can

"no",

least

is most

here - in terms of such

construction

above

application

"yes"

F

to any measurement,

listed

two the

this

prior

as

effects

(6.8),

c

for

a

output

therefore Therefore

implicitly,

single 2,

the

the

output

the origiall

effects

e B.

The

correspondence

mapping F

b,

fixed

~ e B,

all

has been discussed

necessary

already

of

are

F : ~ ÷ F immediately

between

outputs

~ ~ B and

of B onto B c L(H). obvious

from

effects

The properties

its physical

meaning.

F

e B defines

a

of this mapping First,

obvious-

IIi

ly,

the

trivial

outputs

I and

represented by the operators

@

correspond

for mutually

the

trivial

effects

1 and O, respectively:

FI = 1 , Second,

to

exclusive

F~ = 0 .

outputs

a and

(6.10) 8 the probabilities

~ 8 -- "either a or 8" to be "yes" must behave addltlvely; t r ( F a , 8 W)

for

i.e.,

tr(FaW) + tr(FsW)

=

for arbitrary states W, which implies F This

can be easily generalized

= F

+ F8 .

to disjoint

(6.11)

unions

of n > 2 outputs

in

the form F

= F al~

Finally,

since

~Aa'

...W~ a n

+ ... + F ~i

.

(6.12)

an

= ~ and e ~ a' = I for all

a, (6.10) and

(6.11)

imply Fa, = 1 - F a = F'a "

i.e.,

F , coincides

with

the

effect

(6.13)

F'e = "not Fa"

introduced

in

§2,

as expected.

For arbitrary outputs a and 8, (6.12) and the relations a = (aAS)~

(=^8')

8 = (aAS)

~(='A

8)

and

av

B

imply a generalization

=

(a ^ 8 ) ~ ( a ^

8') ~ ( a ' A

8)

of (6.11),

Fav

8 -- Fa + F 8 - F= A B "

(6.14)

112

In particular, we get from this F~v Since F A B both

= 0 means

"yes",

(6.15)

B = F~ + F B

(6.15)

the

more

but not vice versa.

F~ A B

-- 0 .

(6.15)

that the outputs e and B are never found to be

has

is somewhat

if

same

physical

general,

background

as

(6.11),

because e A 8 = @ implies

but

F ~ B = 0

We leave it to the reader to derive a generalized

version of (6.15), F

= F ~i v . . . y e n

Since

all

from

(6.10)

+ ... + F el

properties and

characterize F.

if

F

en of

the mapping

(6.12),

these

(Actually,

= 0

for

i # k .

eiAek F : ~ ÷ F

two

listed

requirements

so far follow

are

sufficient

(6.11) and one of the two equations

to

(6.10)

would also be sufficient.)

The

physical

the

interpretation

instance, only

meaning

the

if

of

of

the

effect

both

the effects

e B follows

corresponding

FIA2

F 1 and

F

is

F 2 are

apparatus

triggered

triggered,

immediately

outputs

case

and

of subsystem

F(1A2,)V(I,A2

)

effects, may

in

be

the

apparatus

b

and

may

therefore

be called

however,

represent which

to

mean

well-defined

would

permit

that

the

operators

calculational

rules

(6.13).

If

F~ and FB.

calculate

there

were

the

and

FI, , F l y 2 ,

"F 1 or F2" ,

"or"

for

is

only

effects

corresponding

true

rules

for

also

as

effect

= "F ~ or F~" directly

This

and

and

This notation should not be misunder"and"

F A B -- "F~ and FB" and F v B

e.g.,

"not FI" ,

"not",

to

if

for the particu-

Similarly,

called

"either F 1 or F2", respectively. stood,

§4.)

~ e B. For

on

"F 1 and F2". (We have already used this terminology, lar

from

used

here

operators,

F,

= "not F ",

and uniquely

"not", for

from

according

"and"

and

to

"or",

their successive application - together with (6.13) - would allow one to calculate the

uniquely

existence

already However,

allows

of one

one to

all

operators

additional calculate

F

rule F vB

as we shall prove later on,

~ B from F I and F 2. would from

(In fact

suffice,

since

F A B and

vice

(6.14) versa.)

the effects F I and F 2 do not in

113

general "and"

uniquely nor

determine

" o r " , when

used

rules for (coexistent)

The

preceding

There

exists

B

(6.5)

yields

a mapping

(6.10)

and

generated to

that B

above,

F

represent

e B. Thus

unique

neither

calculational

the following

necessary

condition

for

of two effects F 1 and F2:

tisfying bra

as

effects

effect operators.

discussion

the coexistence

the remaining

(6.9))

contains

F : ~ E B

(6.12),

by

two

onto

a

+

g B,

of the Boolean

elements set

the effects

under F of the particular

F

B

I

of

and

alge2

(cf.

effects,

F 1 and

F 2 as

elements

sa-

such

(6.16)

images

1 and 2 of B,

respectively.

The

Boolean

generated

algebra

by

two

independent

ments.

Condition

the

any more

F I and that

any

as

at

- when

tation

the

least

suitably together

by assuming

le",

as

theorists

also

as

sufficient

completion

The

choice

more

2,

its

apparatus

condition

extended

is motivated

for

tion

has

ever

by

been

abstract

internal

the

F

the

(6.16)

thus

coexistence may

following

proposed.

such

(6.16)

to of

to mean

apparatus, outputs,

b

(at least "in princip-

FI

condition

and

and sufficient

Second,

-

this interpre-

F 2.

(6.16)

The

set

b.

coexistence

First,

no other

has

simple

a

of

as a coexistent

by the apparatus

arguments.

ele-

that such an apparatus

consider of

an

~ e B as

then be interpreted

as a necessary the

of

constructed We

did

interpreted

the elements

to imply

its

measurement

~ B. We go beyond

(6.16)

of

being

one, not refer-

joint

can be

algebra

structure

coexistence

the

existence

- has

add).

of

for

of the set __{FI,F21 , as realized

tion

the

interpretation

(6.16)

but really

usually

of (6.16)

as

definition

all effects

in

here

specific

original

condition

B occuring

and

imagine

can not only be imagined

effects

I

of a measuring

can

and measures

considered

is thus a purely mathematical

F 2. Nevertheless,

one

which

of

(6.16)

-

existence

is

elements

indeed

ring

B

condicondi-

physical

114

background, ably

and its mathematical form can also be simplified consider-

(see below),

(6.16)

so that it is simply applicable

is satisfied,

one may explicitly

as well.

construct

of an apparatus

b for the joint measurement

all

effects

Last

sion

effects,

criterion

of

F

e B.

(6.16)

but not least,

reproduces

conventional

specific applications

the

quantum

of FI,

when applied

well-established

mechanics.

of (6.16), however,

if

a quantum mechani-

cal model other

Third,

Before

F 2 and

to decl-

commutatlvlty

discussing

such

we shall first transform i t

into an equivalent but much simpler conditon.

In this connection the four particular elements i A 2, of

B,

play

listed a

unique kind:

above

decisive way

I A 2',

under

rSle.

1'1% 2,

(6.6)

Every

as a disjoint

and

excluding

element

e e B

each

can

be

union of n 4 4 elements

n = 0 yields ~ = @; n = 1 yields

selves;

i' I% 2' other

represented

in a

of this particular

the four elements

for n = 2 we get the two elements

palrwlse,

(6.6) them-

(6.7) and the four elements

(6.5) - the latter, because (IA

2) ~ , * ( I A

(IA

2) , ~ ( 1 ' /X 2) :

2')

:

1 A(2

V 2')

= 1A

I

:

1 ,

(6.17) (i V I')

A2

:

I A2

: 2 ,

etc.; for n = 4 we get (I ^ 2) ~ ( I A 2 ' ) :

(I A

(2 V 2 ' ) )

%>(I'A V

(I'A

2) ~ , ( I ' A (2 V 2 ' ) )

2')

: 1Vl'

: I

(6.18)

,

so that, finally, (I A 2) ~ (I A 2') ~ ( I ' A etc.; the

i.e., last

n = 3 yields

equation,

implies B = ~'.)

we

2) = ( l ' a

the remaining

have

used

(6.18)

2')' = 1 V 2

four elements and

the

fact

,

(6.8) of B.

(In

that e ~ ' 8

= I

115

Consider

two effects F 1 and F 2 satisfying

the coexistence

condition

(6.16). Then there exist four effects, FI2 = FI A 2

'

FI2' = F I A 2 '

'

1 (6.19)

FI, 2 = F I , A 2

,

FI'2' = F I ' A 2 '

the images under F of the four elements (6.18),

together

with

(6.10)

and

J

'

(6.6) of B. Eqs.

the additivity

(6.17) and

property

(6.12)

of

the mapping F, imply

FI2 + FI2, = F 1 ,

(6.20)

FI2 + F1, 2 = F 2 ,

and + FI2, + FI, 2 + FI,2, = 1 .

FI2 More generally,

since an arbitrary element = ~ B is a disjoint union

of n ~ 4 elements ~i from ding

effect

mapping

F

(6.21)

is

the

F : B ÷ B

(6.19) are known.

is

(6.6),

(6.12)

of

n

sum

thus

the

completely

Actually

implies

effects specified

that the correspon-

F~i if

from

(6.19).

The

the four effects

it suffices to know three of them,

e.g.,

FI2, FI2 , and FI, 2. They satisfy, because of (6.21) and FI,2, ~ 0,

(6.22)

FI2 + FI2, + F1, 2 < 1 , and the missing fourth effect FI,2, is obtained from (6.21):

(6.23)

FI,2, = i - (FI2 + FI2 , + FI,2) . (Eq.

(6.23)

immediately

implies

FI,2, < I and - with

(6.22) - also

FI,2, ) 0; thus it really defines an effect.)

We

have

according and

thus to

shown

that,

condition

FI, 2 satisfying

if

two

(6.16),

(6.22),

terms of these three effects

effects

there

such

exist

F 1 and three

that F 1 and

F 2 are effects

F 2 may

coexistent FI2 , FI2 ,

be written

in

in the form (6.20). Assuming that, con-

116

versely, with

three

prove

For

two

effects

that

purpose,

(6.21)

FI,2, ,

in

elements

is

shown

(6.6)

These

the

way

operators

as

Eqs.

(6.10)

trivially

effects

they

FI

satisfied

have

and

all

belong

The

F2,

that

of

the

to L(H). and

to

respectively.

As

F

validity whereas

for

then - as ~ ~ B (6.19). arbitrary obtai-

of n ~ 4 diffe-

of

the

is

effects 1 and

indeed,

the

first

the second

(6.12)

elements

Thus,

of the type

F : ~ ÷ F

sums

to

particular

elements

F

so

are > 0, and also ~ 1 by

condition of

FI2

effects

define

the mapping

(6.21),

the

four

particular

The

construction

F also maps

effects

remaining

properties.

additivity

explicit

four

now

(6.23),

F exists,

operators

(6.18)

four

by

a mapping

taken

to show

required

from

(6.20),

be

FI,2,

E L(H)

the

(6.20)

we shall

: ~ ~ B ÷ F

of

the

now

(6.19),

the

and

of

F

form

(6.16).

the

images

If such

under

sums

the

satisfied.

(6.17)

lowing

from

of

F

remains

follows

consequence

a mapping

can

has

therefore

as

F

effect

interpret

(6.19),

(6.22),

condition

a fourth

we

(6.16).

images

it only

this

(6.21);

We

with

representations

in

Then

of B under

-

and

define

satisfied.

in the

FI, 2 satisfying

the coexistence

first

represented

E B,

by

we

in condition

above be

rent

FI2 , FI2 , and

accordance

considered

may

F 1 and F 2 can be represented

that F 1 and F 2 satisfy

this

ned

effects

an

of

one is

immediate

F . Finally,

2 of B into

condition

(6.16)

the is

for F 1 and F 2.

thus

proved

simpler

that

coexistence

Two

effects

if

they

F I and

can

be

condition

(6.16)

is

equivalent

to the

fol-

criterion:

F 2 are

coexistent

represented

in

the

if and only form

(cf.

(6.20)) F 1 = F12 + FI2, in

terms

tisfying

of (cf.

three

,

effects

F 2 = FI2 + F1, 2 FI2 , FI2, and F1, 2 sa-

(6.22)) FI2 + FI2 , + FI, 2 ~ 1 .

(6.24)

117

The

physical

F1,2,

(the

(6.19), above, F2",

interpretation

latter

and

is

being

most

according

of

the

defined

easily

four

by

effects

(6.23))

expressed i n

FI2 , FI2,, F1, 2 and

follows

the

immediately

terminology

from

introduced

to which they may be called "F 1 and F 2 , "F I and not

"F 2 and not FI", and "not F 1 and not F2" = "neither F I nor F2" ,

respectively. single that

If,

therefore,

microsystems single

tr(Fl2W),

by

a

the

and

suitable

microsystems

wheras

FI

in

sense,

measured

W

trigger

for

the

then

operators

the

both

triggering

and

but

In

the correlations

between the results of joint measurements

effects F 1 and F 2. Generalizing

on

probability

FI

F1

F12

together

tr(Fl2,W) , etc.

coexistent

this

are

apparatus,

state

probability

F2

not

to FI,2,

F2

is

F2

is

describe

of the two

a terminology already used

in §4, we therefore call them correlation effects.

The

three

uniquely

correlation

determine

effects

F12 , FI2 , and

all effects

F2

besides

are

given

in

as

F , = s B, and thus,

the two effects F I and F 2 (cf° and

FI,2,

(6.20)).

advance,

it

among

If, as usual,

therefore

also

shown

above,

them,

also

the effects F I

suffices

to

know

them only a single one of these three correlation effects, as

the two others may

then be calculated

from

(6.20).

Since Eqs.

(6.20)

and (6.21) also imply

F 1' = FI, 2 + F1 ,2'

the knowledge

of FI, F 2 and FI,2,

effects

F

Before

drawing

general

criterion

application

of

performed and

(6.25)

,

is also sufficient to calculate all

.

coexistence

denoting

F 2' = FI2, + F1 ,2'

'

two

(6.24)

effect

the selective by

F 2 = ~*G,

f,

the

by

the

aparatuses

we

shall

example

two

effects to

(6.1).

(6.1),

operation,

measured In

first

illustrate the

the

successive

f and g. With ~ and $ = ~ + ~'

and non-selective

according

fects (6.19) are

conclusions,

this

together case,

respectively, are

as

F 1 = F = ~*i

the correlation

ef-

118

FI2 = ~*G

To

show

the

,

this,

consider

apparatus

tr(FW)

FI2 , = ~*G'

f

= tr(~W).

,

FI, 2 = ~'*G

microsystems

-

i.e.,

Those

FI,2,

in a state

the

systems

,

effect

which

have

W.

F1

= ~'*G'

They

-

first

with

triggered

. (6.26)

trigger

probability

f go into

the new

^

state

W = ~W/tr(~W),

i.e.,

the

the

effect

successive

of the effect lities, (We

fled,

F2

-

afterwards

probability

of

both

(6.26)

= tr(G.~W)

presented follow

this

f and

argument

in

Eqs.

g

probability

for

the

-

for

occurrence

of these two probabi-

This §2.)

(6.20)

apparatus

The

g - i.e.,

= tr(~*G.W).

similarly.

the

tr(GW).

"F I and^ F2" = FI2 - is the product

already

in

triggering

with

triggering

tr(~W).tr(GW)

have

tions

thus

implies The

and

FI2 = $*G. ~

remaining

(6.21)

are

equasaris-

since FI2 + FI2 , = ~*(G + G') = ~*I = F = F 1 , FI2 + FI, 2 = (~* + ~'*)G = ~*G = F 2 ,

and FI2 + FI2 , + FI, 2 + FI,2,

= ~*(G + G') + ~'*(G + G') =

(Alternatively, last

three

as remarked

We

shall

show

and

use

Eqs.

effects

in

+

(6.20)

~''i

and

(6.26)

=

F +

F'

(6.21)

from

FI,

=

i

.

to calculate F 2 and

the

FI2 = ~*G,

above.)

derive

from

(6.24)

some

general

results

on

pairs

of

effects.

i__t. Two effects

To

could

correlation

now

coexistent

we

~*i

this,

F 1 and F 2 with F 1 ~ F 2 are coexistent.

take

FI2 , = O.

FI, 2 = F 2 - F 1

F 2 • I), possibility

and of

(6.22)

(the is

choosing

Then,

latter valid

by

(6.20),

being

since

~ 0

we

must

since

set

FI2 = F 1

F 2 ~ FI,

FI2 + FI2 , + FI, 2 = F 2 • I.

FI2 , -- "F 1 and not F2" = 0

means

that,

and The on

a

119

suitable rence

apparatus

of

F 1 is

that apparatus

2__t. Two

always

F 1 and

F 1 < F~

joint measurement

accompanied

FI2 = 0 and

exclude

equivalent

described

To show this,

by

other

(i.e.,

FI2 , = FI, Clearly,

never

each

and

FI2 , = FIF ½ ,

easily

other,

the

checked. same

also for arbitrary

joint

measurement

zed

as

successive

First,

is

effects

The

above:

is

that

thus

the

symme-

F1, 2 = F2;

then,

means

together)

on

indeed,

that an

F1

and

apparatus

= 0 are coexistent.

FI, 2 = F;F 2 ,

are > 0 and < 1 since,

< ilfll~ = (f,lf)

is

a

on

of F 2.

(Note

and

FI2 = 0

occur

[FI,F2]

apply

of

for As

all the

true F

for

= FIF~

.

(6.27)

e.g.,

f E H. four

The validity

operators

arbitrary

of

(6.27)

sums

of

(6.20)

commute

them,

and

and with thus

c B.

F 1 and

measurement, an effect

FI,2,

112 112 fl;2 = IIF1 F 2

(f,FiF2f)

(6.21)

coexistent.

i.e.,

set

operators

~ 0

of F2;

the occurrence

to F I + F 2 < I,

(6.20),

F 1 and F 2 with

FI2 = FIF 2 ,

is

occurrence

the occur-

by this choice.

3__t. Two effects

These

of F 1 and F2,

to F 1 and F2.)

thus,

each

the

F I < F~ are

FI2 + FI2 , + FI, 2 = F 1 + F 2 < I. F2

by

of F 1 implies

F 2 with

is

tric with respect

Set

the

the occurrence

effects

relation

for

F 2 described as

in

apparatus

the

by

(6.27)

example

fl performing

may

(6.1)

be

reali-

considered

the complementa-

ry operations

: W

and

thus

+

measuring

_1/2 _1/2 ~I W~2 '

the

effect

~'

: W

~'1 = FI;

+

after

,i/2 ,1/2 F1 W FI ,

this,

apply

(6.28)

an appara-

120

tus

f2

commute,

measuring

F 2.

Then,

by

(6.1)

and

the combined apparatus measures

the

fact

that

F I and

F2

together F I and

1/2 I/2 F~1/2F2F~1/2 ~*F 2 = ~*F 2 + ~'*F 2 = F 1 F2F 1 +

= (F I + FI)F 2 = F 2 ,

as desired,

while

(6.26) - with G = F 2 and ~,

#' from

(6.28) - imme-

diately leads to (6.27).

The converse of statement 3 is not true, however:

4__t. The

operators

F I and

F2

describing

coexistent

effects

need

not

E

with

commute.

As

an

example,

consider

two

projection

,

F 2 =~E+~-

operators

E

and

[E,E] ~ 0, and set I

F I =~E (Since

.

(f,F2f) = (f,Ef)/2 + (f,Ef)/2 < llfll2 for all

an effect,

f s H, F 2 is also

i.e., ~ 1.) Then we have 1 [F I ,F 2] = ~ [E,E] # 0

but as F 1 ~ F2, F 1 and F 2 are coexistent

Commutativity

is necessary

for

of the two effects considered

5__t. A decision and

only

coexistent ting;

effect if

completion

F12 = EIF 2 ,

coexistence,

however,

if at

least

one

is a decision effect:

E I and an arbitrary

[EI,F2] = 0.

in particular,

by !.

In

B of

this

{EI,F2}

effect

case,

all

F 2 are coexistent operators

are unique

F

and mutually

in

if the

commu'

the correlation effects are

FI2, = EIF i ,

' 2 , F1, 2 = EIF

' 2' • FI, 2 , = EIF

(6.29)

121

To show this,

we first prove a mathematical

Lemma:

be

Let

E

satisfying

Proof:

a

an

llF1--~fll2 = (f,Ff)

i.e.,

A

Since

arbitrary

< (f,Ef)

E(E'g)

= 0

vector

= 0,

then

of

E1

choice

the

and

of

be

and

for

statement

all

F2

the

are

~

coexistent.

FI2 < El, so that,

g E H,

implies

already

coexistent,

correlation

f e H

i.e.,

FE' = F(I - E) = O, which

part

to

operator

F

an

arbitrary

operator

0 < F < E; then EF = FE = F.

Consider

Ff = 0.

projection

statement:

with

also

we

this

F(E'g)

= O,

get

from

FE = F = F* = (FE)* = EF.

follows

from

effects

(cf.

(6.27)).

by

(6.20),

~:

If

[EI,F2]

represents Now

a

assume

E 1 = FI2 + FI2,,

possible E l and F 2 and

(cf.

(6.25))

.

leads to

= FI, 2 ,

with EIE; = EIE 1 = O, we get

(6.30) imply

tions

mutual

As

and

(6.31)

all

follow

from

other

effects

commutativity

a particular

criterion

show

EIF 2 = FI2 , the

then

(6.29),

thus

(6.30)

EIFI, 2 = FI,2E 1 = 0 .

and

= O,

by the lemma,

EIFI, 2 = FI,mE;

Eqs.

get

thus

(6.29)

E i = FI, 2 + FI,2,

from which,

we

and

EIFI2 = FI2E 1 = FI2

Similarly,

Then

F1/2f = O,

and

Then,

Ef = 0.

first

of "ordinary"

of ~,

F

El

commutes

of Eqs.

(6.20)

follows

case

that

and

E B

are

(6.31)

with

(6.29).

(6.21). also

F 2 = FI2 + F1,2,

The other

With

uniquely

the

three four

determined.

equa-

effects Their

as in ~.

we

obtain

the well-known

quantum mechanics:

commensurability

122

6.

Two

decision

[EI,E2] sion

= 0.

effects In

effects

determined

B

is

a

phism,

In

case,

by E 1 and

are

consists

With

the

EaVE

Boolean

algebra,

F

and

only

commuting

= E , which A,

are

V

if

deci-

uniquely

and

' defined

by

~ = E a + EB - EaE B ,

and

if

of mutually

operations

operators

= EaE B ,

coexistent

the

mapping

E'a = 1 - Ea

F : B + B

'

is

(6.32)

a homomor-

i.e.,

first

= EaA

statement

the particular

described

EB

and

case

by projection

,

the

They

are

not

only

i.e.,

EI2EI2 , = 0, onto

considered,

of

the

B ,

all

Ea, = E'a "

F

e B

correlation

(6.33)

follow

effects

from

(6.29)

5. are

operators,

,

mutually etc.,

mutually

E v 8 = EaVE

uniqueness

FI2 = EIE 2 = El2

project

B

E2

operators),

E 2.

projection

EaAB

The

and

(projection

for commuting

EaAEB

this

E1

FI2 , = EIE ~ = El2 , , etc.

commuting

which,

as

orthogonal

but

is

even

well

subspaces

(6.34)

mutually

orthogonal;

known,

means

that

they

of

Furthermore,

Eq.

H.

(6.21),

El2 + El2 , + El, 2 + EI,2,

means

that

H

Since

all

operators

operators are

also

sum of

is

from

the

E

~ B

(6.34),

they

projection

the

direct

ranges

sum of are

sums

likewise

operators

of

the

E

-

four of

= 1 ,

subspaces

n < 4

commute

each

the n projection

E

different among

projects

operators

EI2H to EI,2,H.

from

projection

themselves, onto

the

(6.34)

and

direct

which

add

up to E .

Consider sented

now as

two

such

disjoint

operators,

unions

of

Ea

and

elements

E B. of

When B

from

~ and

~ are

(6.6),

E

repreand

EB

123

become can

analogous

easily

be

sums of the corresponding

seen,

a ~ B is

the

operators

disjoint

union

from

of

(6.34).

those

As

elements

from (6.6) which occur in both a and B, whereas a v B is the disjoint union of those elements representations

of

a

from (6.6) which occur in at least one of the

and

B.

On

the

other

etc., E E B is the sum of those

operators

both

EB,

sums

representing

E

and

and

hand,

from thus

since

(6.34)

in E , E B or both of them,

which

coincides

whereas E a + E B - E E B is the sum of those operators occur

E12EI2 , = 0, occur

with

in

E A B,

from (6.34) which

thus being equal to E v B. (EaE Bhas

to be subtracted from Ea + E~ in order to avoid doublecounting of the operators

(6.34)

which

(6.14).) With A , tors,

we have

V

occur

and

in

both

E

and

E B.

Compare

also

Eq.

' as defined by (6.32) for projection opera-

thus proved

the first two equations

in (6.33), whereas

the last one already follows from (6.13). The preceding argument also yields

the known geometric

defined

by

(6.32): E

E H ~ EBH E

of

AE B = E E B

the

two

+ E B - E E~

V E8 = E

by E H and EBH.

interpretation of the operations projects

subspaces

projects

onto

As is also well known,

onto

E H

the

and

~

and

v

intersection

EBH,

whereas

the subspace E H + EBH E'e = 1 - Ee projects

spanned onto the

orthogonal complement of the subspace E H.

The Boolean algebra and

'. By

with

(6.33),

respect

applied

to

B is closed with respect then,

the

to mutually

to the operations

the set of projection

analogous commuting

operations

projection

operators

defined

by

operators,

A ,v

B is closed (6.32).

these

When

operations

(6.32)

are well known - and easily checked - to satisfy the calcula-

tional

rules

(6.3)

and

(6.4) of a Boolean algebra,

with the rBles of

and I taken by the operators 0 and I, respectively;

(E

A EB)' = i - EaE B = (i - E ) + (I - E B) - (I - E )(I - E B) = E'a + E~ - E'E'~B = E'a V E~

By

(6.10)

algebra

e.g.,

and

(6.33),

structure;

the

i.e.,

it

mapping is

a

F : B + B

.

preserves

homomorphism.

As

the

a % B

Boolean does

not

124

necessarily

imply

E

# E~,

(i.e.,

an isomorphism).

It

is

also

(6.32)

for

"or"

and

well

known

coexistent "not",

the mapping

that

the

decision

F is not

operations effects

respectively.

This

A,

may now

be

and the physical m e a n i n g of the effects

7__7_.In

general,

the

two

set

of

According nor

F2

consider

coexistent effects

pletions

F , ~ ~ B

effects

{FI,F2}

may

F1

are

and

have

and

F2;

several

' defined

interpreted

E

not

one-to-one

"and",

immediately

^~'

E

vB

uniquely

i.e.,

as

by

a

and E ,.

determined

given

different

from

by

coexistent

coexistent

com-

B.

to is

effects

V

follows

(6.33)

the

in general

~,

a

such

decision

non-uniqueness effect.

is

As

a

possible

very

only

if

instructive

neither

example

F1 [I],

the effects 1 FI = ~ E ,

with a "nontrivial"

projection

I E' F2 = ~ E +

operator

E (i.e, E # 0 or I).

i)

!

Since F 2 -- FI, F I and F 2 are coexistent,

FI2 = 0 ,

(6.35)

and we may take

and thus FI2 , = F 1 = ~ E ,

(6.36) FI, 2 = F2 = ~i E + E'

according

to ~.

On the

other hand,

F 1 '2' = 0 ,

,

since

F I < F 2 as well,

we may also

set

F12 , = 0 ,

and thus FI2 = F I = ~ E , (6.37)

FI, 2 = F 2 - F 1 = E'

as in i- Finally,

since

[FI,F2]

,

FI,2,

1 = ~ E ,

= 0, a third possibility

is to take

125

I

I

F12 : FIF 2 = ~ E ,

FI, 2 = F~F 2 = F 22 = ~1 E + E' as

in 3. For

(6.16)

the example

may

be

F : ~ e B + F

e L(H).

-

not

only

for

~ = 1A

Indeed,

in

yielding 2),

but

as can easily

,

with

These

three

mappings

effects

ranges

be proved,

the coexistence

least)

three

different their

therefore, (at

for

the choice

(6.37)and

(6.38)

from

suitable

B = FB c L(H)

are

(6.36)

condition

different

differ

F

i I B : {0, ~ E, ~ E + E', wheras

(6.38)

, 2, = F2FI = ~1 E , = FIF

FI,2,

(6.35),

satisfied

1

FI2 , : FIF2 = ~ E ,

each

other

~ ~ B

(e.g.,

also

yields

mappings

different.

the set

I} ,

lead to

B = {0, ~1 E, E, E' , ~I E + E',

i1

and 1 I 3 1 I 3 B = {0, ~ E, 5 E, ~ E, ~ E + E', ~ E + E', Z E + E',

I}

,

respectively.

As

shown

the of

before,

correlation them

if,

also

given.

this

simply

suits

of

depend

not

lar

the mapping

as

means

joint only

used

uring

apparatus.)

lence

classes

should

and

it

of F

should In view

rather

be too

even

here,

statistical of

effects

the be

FI

suffices

them

realizable

of the fact

by F I and F 2,

of

the

on

between

single

the

re-

microsystems

but also on the particu(Note

by means effects

that,

(6.16), of

according every

a suitable

F represent

apparatuses

correlations

surprising;

on

condition

effect

one

determined

F2

that

specify

F I and F 2 are

together.

coexistence

to

correlations

and

by specifying

the effects

themselves,

than particular

dependence not

the

for measuring of

determined

F is not uniquely

measurements on these

choice

fects

that

interpretation

apparatus

(6.19),

therefore,

sible

ble

is completely

in the case considered

If,

apparatus

to our

effects

F

between

the contrary,

posmeas-

equiva-

f, such possicoexistent

one might

ef-

rather

126

be surprised these

that in particular

correlations

turn

out

cases,

to

be

as specified above in 5 and 6,

independent

of

the

choice

of

the

measuring apparatus.

For

our example

ferent

mappings

choice

(6.36)

possibility and

(6.35), F

listed

of

measuring

ratus.

Then,

F12 = 0,

by

as

apparatus

for

effect

E 2 = E'.

each

other

to

the

together

To

joint

These

FI

the

effect

means

"F 1 or E2"

"either F 1 or E2",

F 1 + E 2 = E/2 + E' = F 2. FI

implies

accordance

the

by

thus

of

-

the correlaions

between

consider

and

the

and

an

decision

they exclude

apparatus

This a p p a r a t u s

since

also meas-

"F 1 and E2" = 0

-

actually

Eq.

(6.15))

definition,

the

occurrence

of

"F I or E2" = F 2 at

this

apparatus,

in

by

F12 , = 0 in

(6.27), of

consequences

i.e.,

(cf.

the choice

remarkable

F1

FI, and

other,

(6.37),

chosen

as

with

application

each

by ~,

coinciding

Moreover,

of

effects

of F 1 at this appa-

F 1 = E/2

(6.29).

of

case

choice

The

mentioned

fl measuring

exclude

the

which

uring F I and F2, respectively,

Some

F2

occurrence

successive

repeatedly

apparatus

the

as a particular the

complementary

two

coexistent,

of

easily.

the

and

are

quite

and

measurement

irrespective

effect

visualized

the three dif-

obvious

realize

"F I and E2" = FIE 2 = EE'/2 = 0, ures

realizing

of F 2 as the non-occurrence

(6.36). the

be

effect

definition,

in

can

an arbitrary

the occurrence

apparatuses

above

corresponds

F 2 = F 1'". take

define

suitable

(6.37).

the choice suitable

(Compare

!.)

with

Finally,

(6.38) may be realized

apparatuses

fl

and

by

f2 meas-

as explained in 3.

of

the possible

coexistent

effects

apparatus

dependence

of

may also be illustrated

by

the example (6.35).

According

to ~,

on a suitable

two effects measuring

another

apparatus

mentary

effects

apparatus does

not

is

F 1 and F 2 with F 1 < F~ exclude

apparatus.

used.

As

an

F I and F 2 given

described exclude,

by but

(6.37): rather

This

need

example,

by (6.35), In

this

implies

not

be so,

consider

the

as measured

case,

the

the occurrence

each other however, two

if

comple-

together by an

occurrence of F 2.

of

FI

(In view

127

of

such

possibilities,

the

effect

F'

the

obvious

it might

by "not F".

and most

appear

However,

"natural"

two

tus !.

effects

the

this,

this

take

described

again by

not

the

be

(6.36),

suggestively

of measuring

F'

together

then on a suitable

the occurrence

true

effects

quite

to denote

(6.35) by the choice (6.36).)

of F 1 implies

need

misleading

expresses

F 1 and F 2 satisfy F 1 ~ F2,

occurrence

Again

this

possibility

with F, as realized in the example

If

a little

for

other

according

apparatuses.

F I and F 2 from

the occurrence

of F2,

(6.35):

of F I excludes

appara-

To

to

exemplify

On an apparatus the occurrence

of

F2, rather than implying it.

Two

identical

to i, this (as

we

effects,

may

case.

choose

On

an

FI2 , = 0),

occur the

output

channels. uring one

an

an

implies

described

versa

by

can

this

be

f measuring

coexistent.

F12 = F and

(as FI, 2 = 0);

apparatus

apparatus

always

choice, i.e.,

realized

F

into

same

effect

apparatus:

corresponding correlation

Consider to

again

(6.37)

effects

but

F by reading

rather

for

FI2

and

effects

their

joint

FI,2,

two

than always

occuring

F 1 implies

by feeding

however, output and

measurement. coincide

output of meas-

channels an

of

apparatus

Then

with

together,

F2

F 2 always

different

(6.35)

both

FI, 2 = 0 in

simply

two different

the

According

F 1 and

There are also less trivial possibilities,

the

F = E/2,

vice

Such

of

are

FI2 , = 0, which

apparatus

and

together.

F 1 = F 2 = F,

the

the

two

effect

they actually

ex-

clude each other.

Such

things

effects

FI

E I < F½,

can and

F 2 < El,

so

that

happen,

F2

E I and

to

not

is

a

F 2 always

therefore

the

however,

decision

E I if F 2 ~ El,

and E 1 implies

lemma

in

~

yields

i.e.,

F2

implies

EIF 2 = F 2 E1.

In

the

each

say

each other.

F 2 if E l < F 2. that,

second

one

by

case,

of

F 1 = E l.

EIF 2

In particular,

exclude

so

least

other: E l < F½

in ~ implies

FI2 = EIF 2 = 0, by (6.29).

sion effects E and E' always

at

effect,

exclude lemma

if

is

the

two

Then,

if

equivalent

(i - EI)F 2 = F2, complementary

Moreover,

In the

deci-

F 2 implies

first

case,

the

(6.29),

FI, 2 = E~F 2 = 0;

E l ~ F2

yields

F½ ~ E~,

128

so

that,

again

by

!

the

F12 , = EIF 2 = 0;

i.e.,

forms

when

together,

measurements of

all

of

these

ment.

Indeed,

spond

to

one

E 1 implies applied

and

the

measurements if

' 2' = F 2' EIF

lemma,

two

measurements

F 2. Finally,

to

must

of

a

same

outputs E,

FI2 , = FI, 2 = EE' = 0 which,

and

single

if some

effect

identical

some

i.e.,

F I = F 2 = E,

two

then

then

per-

or

more

the

results

single

experi-

apparatus

explained,

(6.29),

apparatus

every

2

already

E,

in

I and

as

of

by

microsystem,

decision be

thus,

both

corre-

(6.29)

yields

implies

the desired

conclusion.

The

last-mentioned

I.e.,

if F

apparatus systems

is

not

which in

such

in

that

to

the

a way,

ble

W.

measurements always

preceding

respect,

We of

give

identical

E

turn

like

"properties"

measurements

According

the

vial)

to be of

it

is

that

at

least

effect. following

projection

This

replaced

ger estimate

be

the

F

on single results show

in

obtained

micro-

of

these

replace

correlation

effects

the

first

and

"no"

so in

are non-zero

for suita-

succinctly:

Different

more same

by

an

this,

microsystem

is a decision

itself,

need

effect.

measuring

dependent,

microsystem

not

In this

decision

ef-

and thus appear more than

the

uniqueness

of

results

of

F.

one

E

[FI,F2] here,

for

of

Take

operator

(6.22)

on

unless

sufficient

example.

Since

result

F

condition

0 < ~ < ~ < I. may

this

exists

FI2 , = FI, 2 = F - F 2 # 0,

"yes"

less apparatus

!,

the

effects.

there

the To

by tr(Fl2,W),

results

the

other.

yields

effect

of other effects

to

F : ~ ÷ F

by

out

the

each

obtaining

results

decision

then

sometimes

a n d choose

now

as given

same

(F 2 ~ F),

least

argument,

for

for

two F measurements

from

express

the

fects

at

this

may

therefore,

decision

that

measurement,

states

effect

different

(6.27);

characteristic

together

probabilities

the second

is

decision

are

the

according

a

performs

two measurements E by F

property

is

the

the not

F 1 = ~E and

real

two

necessary, and

(6.20)

FI

F 2 are

~

mapping

and

F2

is

however,

as

shown

F 2 = BE'

numbers

= 0, F I and using

effects

the

with and

~

coexistent.

and FI2 > 0,

a

a

(nontri-

satisfying Condition

by the

stron-

129

(6.39)

FI2 + FI2, + FI, 2 < F I + F 2 = ~E + BE' < B(E + E') : ~I

The

effects

G I = FI/B = ~E/B

and by (6.20)

and

G 2 = F2/B = E'

they may be written

G 1 = GI2 + GI2, with correlation

are

also

coexistent,

in the form

,

G 2 = GI2 + GI, 2

effects GI2 = FI2/B , etc., which indeed satisfy

G12 + GI2 , + GI, 2 < 1 , according

to

(6.39).

correlation

effect

correlation

effect

But G12

because is

G 2 = E'

unique

F12 = BG12

is

(cf.

is

also

a

decision

6).

Therefore

unique,

which

effect, the

the

original

implies

unique-

ness of F.

8. Consider,

as in §4,

a composite

acting

subsystems

effects

(cf. (4.10)),

are

coexistent,

correlation

I

and

system

II.

Then

consisting

arbitrary

of two noninter-

pairs

~i = FI @ III '

~2 = II ~ FII '

according

and

to

3,

a

of

subsystem

(6.40)

possible

choice

is

the

FII only

(and

FII

thus

possible

F : B ÷ B C L(H). (6.41)

decision

therefore

most

subsystem

effects

Fl

(6.41)

or F2 ) is a decision

choice,

Moreover,

is in a certain

is a

the

effect ~12 = "El and ~2" is, by (6.27),

~12 = ~I~2 = FI ~ FI! "

If F I or

for

"natural" ~I

and

leading

as discussed

sense

effect,

thus

it

possibility ~2"

(See

there occurs as Eq. (4.9) - in §4.)

to

in detail

the "natural"

since

effect, a

then

unique

choice

one also if neither

F I nor

to

the

simplest

for the joint measurement the

mapping

the

corresponds

in §4,

(6.41)

derivation

of

(6.41)

and

of the - which

130

Nevertheless,

there

are

also

pairs

which (6.41) does not represent

of

subsystem

effects

(6.40)

for

the only possible choice. As an examp-

le, set

I ~2 = II @ 2 Ell

~I = ~1 E1 @ Ell ' with two nontrivial

projection

operators

E I and Eli . In this case, we

may take F = ~(E I @ Ell ) -12 with

an

arbitary

real

~12' = ~i - ~12

and

~

between

0

and

~i'2 = ~2 - ~12;

1/2.

Eqs.

these

two

(6.20)

then yield

operators

really

belong to L(H) since, e.g., i ! ) ~I ~ ~I - f12 = f12' = 2(El ~ III) - a(El ~ Eli) i I ' ) ~ 0 ~((m I @ iii) - (E I ~ Ell)) = ~(E I ~ Eli Moreover,

since both ~I and ~2 are < !/2, we have

~12 + ~12' + ~i'2 = ~i + ~2 - ~12 < ~1 + ~2 ~ ! , so that (6.22) is also satisfied.

Besides

illustrating

between

subsystem

may

exist

pair

of

Except

coexistent

"natural" how

effects,

infinitely

for

the

choice

apparatuses

the possible the

many

(6.41)

above

different

effects,

particular

non-uniqueness

as case

example mappings

parametrized ~ = I/4

of

also

here

- which

all

these

mappings

F

correlations

shows

F : B + B by

that for

the

unknown,

would

look

there

a given

number

corresponds

for ~12 - it is completely

realizing

the

to

~. the

however, in prac-

tice.

We

conclude

effects

our

discussion

of

the

particular

F I and F 2 with the construction

case

of two coexistent

of a quantum mechanical model

131

for

their

joint measurement.

bed

as

quantum

mechanical

system

and

assumed

form

state

a Wa,

is

binary

scattering

tor

on

S

--

output have

tors

rather

"read"

microsystem. performed

now

a

be

This

at

Ela and

than the

with

the only

thus

as described

E 2a on

the model

with

H a.

state the

space

by a unitary

a single

is

its of

is descriand

scattering to

yes-no

yes-no

pair

"reading"

is

a

opera-

have

interaction

two

initial

considered

Their

by a coexistent the

Ha

assumed

one.

after

consists

Actually

apparatus

microsystem

apparatus

apparatus

"reading"

together,

in §5,

characterized

H = H @ H . But

channels, to

to

system,

--

As

two

outputs with

the

measurements

of effect performed

operaon

the

composite system, and is therefore described by the two effects a a I ~ E 1 and I @ E 2 on H. As is obvious, e.g., from the condition a a (6.24), the latter are indeed coexistent if E l and E 2 are. (The a

converse sion

is equally

effects.

assume

this

obvious

Although,

as

- e.g.,

from 6 -

notation already a a the moment E I and E 2 may

later

on,

at

apparatus

of

this

a

if E l and E 2 are

our

suggests, still

deci-

we

will

be arbitrary

effects.)

A

model

F 2 of

the

microsystem,

kind

defined

measures

together

implicitly

but

two

effects

uniquely

by

F 1 and

the

equa-

tions tr(FiW)

with

W ~ K(H)

right

hand

after

the

W.)

arbitrary.

side

of

(6.20),

the

(Compare

(6.42)

interaction

As expected,

represent

= tr(l @ E~)S(W @ Wa)S*) ,

of

F 1 and

coexistent

the apparatus F 2 are

remember

for output

with microsystems

coexistent.

"output"

and

In o r d e r E al

effects

and

(6.42)

that

i to be "yes" in the state

to prove a E2,

the

in

this,

we

analogy

to

in the form

four

lations

(5.1),

is the probability

a a a E 1 = El2 + El2, with

Eq.

i = i or 2 ,

effects

between

a

,

a

a

E 2 = El2 + El, 2 ,

(6.43)

E? • (i = I or i' , j = 2 or 2') describing the correlj the two "output" effects E al and E 2a and satisfying,

132

in analogy to (6.21),

a + El2 a , + El, a 2 + E ~ '2' = la . El2

Corresponding

effects

F.. lj

of

the

microsystem

can

(6.44)

then

be

defined,

analogous to (6.42), by

tr(FijW) = tr((l @ E~j)S(W @ Wa)S*) ,

These

i = 1 or i', j = 2 or 2'.

(6.45)

effects

F.. ij have to be interpreted as correlation effects for F I and F 2 (cf. (6.19)) since, e.g., with i = 1 and j = 2', the right hand

side

apparatus -

by

of

(6.45)

effect

definition

for

the

occurrence

a , = ,,Ka Ka,, after El2 -i and not -2

the

interaction,

of

is

the

occurrence

of the effect

FI, F 2 and

the

(6.43)

the

probability

effects

F I and

F2 -

is

"F I and not F2". Eqs.

correlation

effects

Fij

and (6.44) and the definitions

are

equivalent

of

the

which to

the

(6.20) and (6.21) for

easily

(6.42) and

derived

(6.45).

from Eqs.

(For instan-

ce, (6.42), (6.43) and (6.45) lead to a

tr((F12 + FI2,)W) = tr((l @ [El2 + EI2,])S*) = tr(FiW)

for all W, which implies F 1 = FI2 + FI2,.)

Now consider, stent

conversely,

effects,

an arbitrarily given pair {FI,F2}

and a representation

(6.20)

given - but, in cases of nonuniqueness, tion effects and

the

apparatus

Fij.

given of

the

sion,

this amounts

state

W a g K(Ha),

E?lj Eqs.

e L(Ha) (6.45)

type

effects

considered.

to proving a unitary

(i = I or I' are

of F 1 and F 2 in terms of

deliberately chosen - correla-

We will prove that the joint measurement

correlation

the existence operator

with

of FI, F 2

F.. can be "realized" by a model 13 According to the previous discus-

_S on

j = 2 or 2')

satisfied

of coexi-

the

of a Hilbert H 8 Ha,

and

space Ha, a four

satisfying

(6.44),

given

correlation

effects

such

that

effects

F.. e L(H) and arbitrary states W ~ K(H). With coexistent "output" lJ effects E al and E 2a defined by (6.43), then, Eqs. (6.45) and (6.20)

133

imply

(6.42),

so that the apparatus

correlations

described

this apparatus effects F

We

by

indeed measures F 1 and F2, with

FI2 ' FI2,,

etc.

As

remarked

previously,

then also measures - at least implicitly - all other

c B.

shall assume

are d e c i s i o n

that,

effects.

as already suggested by the notation, Then ( 6 . 4 4 )

means

that

the

corresponding

the E?. ij projeca

tion operators project onto four mutually orthogonal subspaces EijHa, and H a is the direct sum of the latter. Moreover, a a commuting projection

operators

a

E 1 a n d E2,

aa

a n d we g e t

aa,

a

E l 2 = E1E 2 ,

(6.43) defines two

E l 2 , = E1E 2

,

etc.,

in accordance with our previous results

(see 6) and with "conventio-

nal"

operations

quantum

mechanics.

Choosing

four

~ij (i = 1 or I',

j = 2 or 2') with ~jl

but

arbitrary

otherwise,

= Fij

we replace

(6.46)

(6.45) by the stronger

require-

ments a

(6.47)

~ijW = Tra((l @ Eij)S(W @ Wa)S*) .

(By

taking

virtue

of

the

trace,

(6.46).

(6.47)

Being

is

analogous

easily to

seen

(5.14),

to

imply

(6.47)

(6.45),

means

that

by the

operation ~ij is performed by selecting those microsystems which have triggered

the

effect

E~. ij

at

the

apparatus

-

i.e.,

the

correlation

effect Fij.)

We are

thus left with

the problem of representing

tions ~ij satisfying, by ( 6 . 4 6 ) a n d

(~2 + ~2' in the form

(6.47),

+ ~'2

four given opera-

(6.21),

+ ~ ' 2 ')I = 1 ,

(6.48)

with suitable Wa, S, and four projection opera-

134 a

tors

E.. satisfying (6.44). An analogous problem has already been ij solved in §5 i n t h e p r o o f of Theorem 2. There two g i v e n o p e r a t i o n s ~

and ~' satisfying (¢* + ¢'*)I = 1 ,

in analogy to (6.48), were represented in a form analogous to (6.47) (cf. (5.20) and (5.21)),

~(')W = Tr a ((I ~ E('))S(W @ Wa)S*) a ' with two projection operators E a and E'a which satisfy the condition E a + E ' a = ia analogous cit

to (6.44). As can be easily seen by inspection,

construction

generalized

of

Ha,

immediately

Wa,

_S, E a

and

E'a described

to the present problem.

in

the expli§5

can

This establishes

be the

existence of the desired quantum mechanical model.

The model as

apparatus

already

remarked,

can also be "used" the operations

to perform operations;

~ij'

by

selecting

e.g.,

the microsy-

stems which have triggered the correlation effects Fij; or the operations ~i = ~12 + ~12' by selecting

and

~2 = ~12 + ~I'2 '

the microsystems which have triggered the effects F I or

F2, respectively; or the non-selective operation

= ~12 + ~12' + ~I'2 + ~i'2' For a given pair of coexistent effects FI, F 2 and given correlation effects

Fij , the choice

still highly arbitrary. tions like ~I' ~2 and ~.

of the four operations The same,

therefore,

~ij

(cf.

(6.46))

is

is true also for opera-

135

The

coexistence

arbitrary

criterion

(6.16)

sets of effects,

A set C c L(H)

may

be

generalized

immediately

to

as follows:

is coexistent

if and only if there

exist a Boolean algebra B and a mapping F

satisfying

: a

(6.10)

B

c

and

+

F

(6.49)

e L(H)

(6.12),

such

that C is con-

flained in the range B = FB = {Fa I a e 2} of F.

Namely,

if C is coexistent,

measurement lar

case

output

there

of all effects

outputs

a

effects

F

form

until a

Conversely, existence

if

apparatus

one

Boolean

c FB ~ C.

algebra

is

arrives

B,

of such an apparatus

b,

above

at

an

and which

(6.49)

satisfied,

c for the joint for the particu-

c can be extended

finally

Therefore

(6.49)

an apparatus

F ~ C. As described

C = {FI,F2} , this

channels,

exists

is we

apparatus

measures

necessary may

by adding

(6.49)

all

coexistence.

consistently

thus taking

b whose

together

for

new

assume

the

also as a suffi-

cient condition.

If

taken

and

the

since here

literally, Boolean

algebra

apparently can have

finitely

of

outputs

of

this

outputs

discussed

bra

As

we

physically

the

With

C,

apparatus before see

condition

called

-

as

in

then to

v

of

(6.49).

a

-

effects

particular

B

' to

the

Boolean

measuring

finitely

case

also

case

also

many

of

"output"

two

alge-

simple

and

infinite

(6.49).

B = FB ~ C also is

C

elements,

together

criterion

set

the successive

the

are

the

type considered

and

there

of

the

in

if

many

only,

as

finite

however,

Thus

c of

and

leads

only

finitely

channels

the coexistence

set

the

of

apparatus

A,

below,

to apply

consist

output

also

seem

possibilities

the

coexistence

of C.

both

c -

satisfying

obviously,

completion

many

interesting

here

B

operations

shall

sets of effects

arguments

a real measuring

application

2"

these

satisfies

coexistent,

C = {FI,F2}

-

a

and

the is

coexistent

136

An arbitrary satisfies since

pair

of effects

condition

{FI,F2}

(6.24),

{FI,F2} ~ C ~ ~_,

and

by

chosen from a coexistent

is

(6.49),

therefore there

are

coexistent. two

set

Indeed,

elements

of

B

1 and 2, say - which are mapped by F into F 1 and F2, respectively. a Boolean algebra,

B contains

ments

(6.6) satisfying Eqs.

etc.

defined

-

i.e.,

by

condition

(6.10)

and

(6.49)

implies

the

(6.19),

coexistence

we

obtain

of

the

mapping

(6.24)

for

the particular

(6.16)

condition

equivalent

then, -

hand,

from •(6.17), F.

trivially (6.49)

coexistence

i and 2 also

As

the four ele-

(6.17) and (6.18). With effects F12 , F12 ,

(6.24)

(6.12)

other

along with

C

is

This

really (6.16)

(6.20)

(6.18)

and

argument

and

the

(6.49)

also

in

shows

(6.24)

that

Since,

this

a generalization and

(6.21)

properties

case C = {FI,F2}.

implies

conditions

Eqs.

case, of

the

the

for pairs

on

two

of ef-

fects.

We do not know whether, provided of

conversely,

a set C of effects is coexistent,

this is true for all pairs {FI,F2} from C. But if C consists

decision

effects

only,

this

can

indeed

be

proved.

Or,

in other

words: A set C of decision effects is coexistent if and only if every pair {EI,E2}

from C is coexistent - i.e., according t o 6: if and only

if C consists of pairwise commuting projection operators.

The

"only if" part of this

der,

therefore,

operators. A,

V

-

(6.32).

stions, limit

of i.e.,

commuting

to these operators

commuting

a Boolean

algebra - with

generated

which

projection

the operations

like,

again

E l... E n E B

and

the

projection

operators,

respect

to these

which

is

operations

B is the unique Boolean algebra of projection

by C.

would

elements

introduce

applying

mutually

In other words,

operators

set C of mutually

Consi-

' as defined by (6.32), we obtain from C a set B ~ C, also

consisting closed

an arbitrary

By successively

and

statement has already been proved.

arise e.g.,

(We leave aside here if B were assumed

some topological

que-

to contain also certain

n÷=limEI A . . . A En with E l, E 2 ... g B.)

notation

EI~¢~...~

E.l A E.3 ~ E.E.1 3 = 0

vious generalization of (6.4) yields

for

En

i @ j.

for For

ElY such

We

...V

En

if

El,

an

ob-

137

E 1A

(E 2 V . . . V E n) = (E 1 A

E 2) V . . . V

(E 1 A

E n) = 0 ,

so that, by (6.32), EI V

(E2 V

...V

E n) = E 1 + ( E 2 V

...V

E n) .

Repeating this argument, we thus obtain

E1 ~

In order we

now

to prove

identify

...~

E n = E l + ... + E n

that C satisfies

the

(abstract)

(6.50)

the coexistence

Boolean

algebra B

condition occuring

(6.49),

in (6.49)

with the above Boolean algebra B of projection operators generated by C,

and

take

for F

the

identity mapping.

definition.

Moreover,

since

in B

represented

by the operators

Finally,

tivity

is

property

(6.12)

the abstract

1 and 0,

is trivially satisfied.

Then FB E B contains elements

respectively,

I and

condition

C,

by

~ are (6.10)

for the identity mapping F the addi-

identical

with

(6.50).

Thus,

indeed,

the

set C is coexistent.

Consider,

more generally,

a coexistent

subset C

of decision effects,

set C of effects

containing a

a Boolean algebra B and a mapping F as

O

in

(6.49).

Since FB = B

B' of B consisting effects,

F

Boolean

F

Co,

there

exists

a nonempty

subset

of those elements a which are mapped into decision

= E . With arbitrary ~ and B s B'

E B are coexistent a

contains

- i.e.,

algebra,

commuting

the

set

the two effects E

projection

B = FB

operators.

contains

also

and

Since B is the

B = "E~ and EB" , F v B = "E~ or EB" and F , = "not E~".

effects According

to !, the latter are again decision effects, and are given by

E E~V

=EE

(6.32)

and

AE

B = E~ + E B - E E B E , = 1 - E

(cf.

=E

(6.33)).

, = E~V

I

EB

(6.51)

= E'

Therefore

B'

contains

along

with

~

and

138

also

~ A ~, ~ V B and

Moreover,

by

(6.51),

coexistent

decision

is closed,

and

tions

A,

mapping

V

~'"' i . e . ,

and

F to B'

effects,

above,

As

algebra,

cular,

C

contains

coexistent jection B

B'

operators

itself

(6.32),

with and

B.

--

of

projection

operators

respect

to the opera-

the

restriction

of

-

the

of B' onto B'. As in the particular

contains

leads

A, V

and

along

with

B of C contains

to the familiar physi' for arbitrary

projec-

generated

is a possible

C

the Boolean

algebra

of proo as shown before,

by C. On the other hand,

choice

for B; hence

B

O

B

is the minimal

coexi-

O

stent completion

The concept

of a given coexistent

of an observable

tions

of

notion

appears

quantum

presenting

discussion

with

defined

operationally

here

An easily which

a

after

the application

very

few

-

such

simple

if

in

terms

indicates

any

-

of

of

since

by

coexi-

the

in §I, observables

are

apparatuses

measuring

apparatus,

an appara-

specified

be an apparatus

"measured

mechanical

thus we

this

by the "clas-

and application.

the apparatus

quantum

apparatuses;

the

especially

or an effect

would

here,

rSle in this connection.

is also completely

example

presented

discussed

of its construction

visualizable pointer

ideas

in some formula-

one. We shall conclude

of observables,

instrument

an observable

sical" description

approach

as a derived

the general

a preparing

tus measuring

the

play an important

In accordance

Like

In

on the contrary

a short

set C of decision effects.

is taken as fundamental

mechanics.

stent sets of effects

them.

of

the whole Boolean o operators generated by C . If thus, in partio decision effects (i.e., C = Co), then every

only

completion

subalgebra

of B' under F - consisting

this homomorphism

of projection

o

by

Boolean

commuting

of the operations

in B'.

Boolean

a

algebra,

is a homomorphism

tion operators

algebra B

i.e.,

' defined

cal interpretation

is

the image B' = FB'

is thus a Boolean

case 6 discussed

a

B'

o

shall

value"

with a scale,

of

the

to a microsystem. observables only

assume

are here

on

observable In practice measured

by

that somehow

139

each

application

yields the

of

the measuring

a well-defined

errors

real number

connected

these measured

apparatus

with

values

any

real

measurement, infinite

idealization.

We leave aside here

idealizations

of

of observables

would become very cumbersome

By

definition,

possible them

measured

it may

values, fied

tageous

also

here,

coordinates

case

are

fore represent

Consider for

of

not

measured measured

"unbounded" components

Borel

sets

on

single

a

no measurement operator

(see (cf.,

F . We

thereby

represents

value

in the Set ~, when

state

W.

in

line

Ch.

like,

the

universal

e.g.,

whose

and which

question

operators thus

of

an

intervals;

more

general

observable

wether

or

F

plete quantum mechanical

e

specify W,

description

and

a

given

into

probability

of

~ -

the in

or,

a yes-

this

the

in other

of the observable.

a

measured

is measured

sense

by an

By definition

finding

"statistics"

the

observable,

L(H).

considered

for arbitrary

states

for

sets

the observable

completely

arbitrary

obtain,

is

not

and is thus to be described

Borel

the

there-

observables.

the

If

position

possible meas-

I, e.g.,

here

I).

a

in the subsequent

interval,

choose

is advan-

I as

include

line

It

system,

tr(F W)

observable

line.

on the given

of

-

thus be identi-

in a given Borel set ~ constitutes

F : ~ + F

F

occur

(i.e., bounded)

[13],

all

besides

is contained

mapping

mapping

e.g.,

contains

as measured

real

the real

microsystem,

certain

The

also

we

description

never

of a particle,

below)

of justifying

but

observables

~ of

an

considered,

real

to a finite

subsets

reasons

value

F ,

we

problem

and could

on the

the whole

Thereby

confined

now suitable

of

effect

to use

that

is clearly

apparatus

which

is finite,

interval

assumption

or even impossible.

observable

numbers

the

In view of

a mathematical

measuring

limiting cases of "real"

technical

class

finite

or momentum

values

the

scale

observables.

the

a

real

every

however,

for all

of

contain

a suitable

discussion

ured

values

of

which

microsystem

value".

precision

the difficult

without

"scale"

in practice

with

scale

the

kind,

single

as the "measured

are given with

this

to a

of

in the

words, the

provide

the

given a

com-

140

In particular,

since

the measured

value will

always

lie on the real

line I and never in the empty set ~, we immediately get

FI = 1 ,

Moreover,

clearly,

measured joint

values

the

in

probability

the

union

sets al... a n (i.e.,

corresponding

F~ = 0 .

finding,

kJi e i of

~ir~j

probabilities

of

(6.52)

in a given

finitely

many

state

mutually

W,

dis-

= ~ for i # j), is the sum of the

for these sets ~i''" an separately;

tr(F~J~iW) = ~i tr(F~iW)

i.e.,

.

(6.53)

i Being true for arbitrary W, this implies the additivity property

F~i

~. F~. 1 i

i of

the mapping

F : ~ ÷ F . In

analogy

to our previous

as

~i"

~i

~i F~ i' here

We

have

thereby

sequences

also

(6.54) we have written,

the "disjoint

omitted that

and

an

Eqs.

First,

disjoint

Borel

upper

limit

(6.53)

and

sets ~i"

in

~i

(6.54)

a union of countably many (arbitrary,

Borel sets ~i is again a Borel set;

hand

(6.53)

of

n

and

(6.54)

exist

also

~i

are

and

assumed infinite

This deserves

rily disjoint) sides

in

union" of the sets ~i

not only for finite but also for countably

of mutually

explanation.

notation,

indicating

to be valid

(6.53)

(6.54)

a little

not necessa-

therefore the left

for infinitely

many

sets

~.. Moreover, in this case, the right hand side of (6.54) exists as i an ultraweak limit. To show this, consider the finite disjoint unions " ~n = ~Ji

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