Quantum Information III Edited by
T. Hida •« A K. Saito
Quantum Information III
Proceedings of the Third Internatio...
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Quantum Information III Edited by
T. Hida •« A K. Saito
Quantum Information III
Proceedings of the Third International Conference
Quantum Information III Meijo University, Japan
7-10 March 2000
Edited by
T. Hida & K. Saito Meijo University Japan
V f e World Scientific wB
Singapore • New Jersey 'London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data International Conference on Quantum Information (3rd ; 2000 ; Meijo University) Quantum information III: proceedings of the third international conference, Meijo University, Japan, 7-10 March 2000 / edited by T. Hida & K. Saito. p. cm. ISBN 9810245270 1. Quantum computers-Congresses. I. Title: Quantum information 3. II. Title: Quantum information three. III. Hida, Takeyuki, 1927- . IV. Sait6, K. (Kimiaki), 1959Title. QA76.889 .1535 2000 004.1-dc21
'
2001026181
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
Printed in Singapore by World Scientific Printers
V
Preface The Third International Conference on Quantum Information was held at Meijo University in Nagoya, Japan, March 7-10, 2000. This volume contains the papers of invited lectures and contributed talks at this conference. The editors are pleased to accept all the papers for publication in this volume at the suggestion of the referees so that this volume is most valuable. The following topics were discussed at the conference: 1) Complexity in Quantum System 2) Quantum Stochastic Processes, Quantum Stochastic Analysis 3) Quantum Computation 4) White Noise Theory 5) Infinite Dimensional Stochastic Analysis 6) Variational Calculus 7) Random Fields 8) Time Reversal Symmetry of Fluctuation This conference was supported by the Research Project "Quantum Information Theoretical Approach to Life Science" for the Academic Frontier in Science promoted by the Ministry of Education in Japan and was also supported by Meijo University. We would like to express our sincere thanks to the Faculty of Science and Technology of Meijo University for their assistance during the conference.
December 30, 2000 Takeyuki Hida Kimiaki Saito Meijo University
Contents Preface A Generalization of Grover's Algorithm L. Accardi and R. Sabbadini Characterization of Product Measures by an Integrability Condition
1
21
N. Asai Tunneling Effect Based on the Nelson's Quantum Stochastic Process Approach — Comparison with a Neutron Spin Echo Experiment T. Hashimoto and T. Tomomura
35
A Quantum Aspect of Asymptotic Spectral Analysis of Large Hamming Graphs Y. Hashimoto, N. Obata and N. Tabei
45
Elemental Random Variables in White Noise Theory: Beyond Reductionism T. Hida and Si Si
59
Gibbs State, Quadratic Embedding, and Central Limit Theorem on Large Graphs A. Hora
67
Algebraic Stochastic Differential Equations and a Fubini Theorem for Symmetrised Double Quantum Stochastic Product Integrals R. L. Hudson
75
Growth Functions for Generalized Functions on White Noise Space H.-H. Kuo
89
Some Properties of a Random Field Derived by Variational Calculus K. S. Lee
97
A Stochastic Expression of a Semi-Group Generated by the Levy Laplacian K. Nishi, K. Saito and A. H. Tsoi
105
VIM
On the Design of Efficient Quantum Algorithms T. Nishino Phylogenetic Relation of HIV-1 in the V3 Region by Information Measure T. Nishioka, K. Sato and M. Ohya Quantum Logical Gate Based on Electron Spin Resonance M. Ohya, I. V. Volovich and N. Watanabe Current Fluctuations in Nonequilibrium Steady States for a One-Dimensional Lattice Conductor S. Tasaki Representations and Transformations of Gaussian Random Fields Si Si Time Reversal Symmetry of Fluctuation in Equilibrium and Nonequilibrium States M. Yamanoi and F. Oosawa Stochastic Convergence of Superdiffusion in a Superdiffusive Medium /. Doku
119
127
143
157
177
185
197
Quantum Information III, pp. 1-20 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company
1
A Generalization of Grover's Algorithm LUIGI A C C A R D I R U B E N SABBADINI
Centro Vito Volterra Universita degli Studi di R o m a "Tor Vergata" Via Orazio Raimondo, 00173 R o m a , Italia
ABSTRACT We investigate the necessary and sufficient conditions in order that a unitary operator can amplify a pre-assigned component relative to a particular basis of a generic vector at the expence of the other components. This leads to a general method which allows, given a vector and one of its components we want to amplify, to choose the optimal unitary operator which realizes that goal. Grover's quantum algorithm is shown to be a particular case of our general method. However the general structure of the unitary we find is remarkably similar to that of Grover's one: a sign flip of one component combined with a reflection with respect to a vector. In Grover's case this vector is fixed; in our case it depends on a parameter and this allows optimization.
1
Unitary operators which increase the probability of the |0> component of a pre-assigned vector
Let \i > (i = 0 , . . . , N — 1) be an orthonormal basis of R . The mathematical core of Grover's algorithm is the construction of a unitary operator U which
2
increases the probability of one of the components of a given unit vector, in the given basis, at the expence of the remaining ones. The necessity of such an amplification of probabilities arises in several problems of quantum computation. For example in the Ohya-Masuda [4] quantum SAT algorithm such a problem arises. In a recent interesting paper Ohya and Volovich have proposed a new method of amplification, based on non linear chaotic dynamics [14]. In the present paper we begin to study the following problem: is it possible to extend Grover's algorithm so that it becomes applicable to a more general class of initial vectors, for example those wich arise in the Ohya-Masuda algorithm? A preliminary step to solve this problem is to determine the most general unitary operator which performs the same task of Grover's operator. This is done in Theorem (1.1) below. The result is rather surprising: we find that, up to the choice of four ±1 (phases), there exists exactly one class of such unitary operators, labeled by an arbitrary parameter in the interval [0,1]. Moreover these unitaries can be written in a form similar to Grover's one, i.e. a reflection with respect to a given unit vector possibly preceeded by a sign flip of one component combined with, where the unit vector in question depends on this parameter in [0,1]. The free parameter in our problem allows to solve a new problem, which could not be formulated within the framework of Grover's explicit construction, namely the optimization problem with respect to the given parameter. We prove that, even in the case of Grover's original algorithm, this additional freedom allows to speed up considerably the amplification procedure. In a forthcoming paper [15] we plan to apply the present method to the OhyaMasuda algorithm. Since an operator U is unitary if and only if it leaves unaltered the scalar products of vectors with real components in a given basis, we shall restrict our attention to unitary operators with real coefficients in a given basis (as the Grover's ones). This restrictes the problem to R^.
T H E O R E M 1 Given the linear functionals: N-l
TJ : a = (a,-) e R ^ n 77(a) = ^ jy.-a; «=o
(1)
N-l
c: a= (aj) € R W
H->
c(a) = J ^ 7;a,t'=0
(2)
3
with 7; and rji real and £1, £2 G {±l}i necessary and sufficient condition for the operator U, defined by: U ^ a , | i > = £ ! ( a o + ??(a))|0 > +£ 2 ^ (a t -+ c(a)) |i >
(3)
to be unitary is that there exist a real number fj0 such that: IADI < 1
70 = £ 5
(5)
7F^T
i* =—jvrr %
(4)
l
^°
= - l + e4/3o
??; = £370
i ^ 0
(6) (7) (8)
where e 3 , £4, £5 are arbitrarily chosen in the set {±1}. P R O O F In finite dimension unitarity is equivalent to isometry. Therefore U is unitary if and only if, for every \a > = J ^ o * a«H > * n e following isometricy condition is satisfied:
where we write rj, c for r)(a), c(a). This condition can be written in the form: 7?2 + 2aO7? + ( 7 V - l ) c 2 + 2 c ^ a , = 0
(9)
With the notation: 7(a)
= 7:=(Af-l)c2 + 2c^aI
(10)
Equation (9) is equivalent to: r,2 + 2a07? + 7 = 0
(11)
4
and its possible solutions are: »/(«) = V = - « o + £4y a o - 7(«)
(12)
Given (12) the funtional r)(a) will be linear if and only if Vetoi - • •, aN'-
«o-7(«)=
E M
(13)
for some real numbers fy indipendent of a. Since the functional c(a) is linear and given by (2), because of (12) and (13), condition (9) becomes:
-a 2 + {N - 1) I J2 Ha, \ i
-a
/
+ I E Pi«i J + 2 E 7i«> E «•• = \ i
/
i
«^o
o + 2 E TjaJ E ^ + E l(N - lhaj + PiPi\ aia3 = o
or equivalently: [ 2 7, + ( ^ - 1)7.7, + A &] *i*3 +
«o [ ( # - l)7o + & - l] + E +2 E [7o + (N-
l)7o7i + A>A] «oa, = 0
(14)
«>to
The identity (14) holds Veto? • • •, AJV> if and only if: (jV-l)7o2 + / ? 2 - l = 0 2 7 i + (TV - 1 ) 7 J 7 J + A/3, = 0 2
Vi.j^O
2
27.- + (N - 1) 7 + A = 0 7o + (N - l)7o7.- + Mi
(15)
= 0
V« / 0 VzV 0
i^j
(16)
(17) (18)
Equation (15) and the reality condition on rj imply that (4) and (5) hold. From (18) we deduce that, for i ^ 0: 7,
= _7o + /3oA 7 o ( N - 1)
(19)
5 a n d , replacing this into (17), we find: 2(7o + PoPi) , (7o + PoPi? 70(iV-l)
+
ll{N-\)
, B2
+ P
=
n
/ l - PI = sin 0
(24)
a n d therefore, from (5):
i.e. t h e p a r a m e t e r s 0O and 70 live o n t o an ellipse in t h e (/? 0 , 7 0 )-plane. W i t h these n o t a t i o n s one has: SZTt \)
77(a) = ( - 1 + e3e4cos
0) a0 + e 3 £ 4 ^—= J2 a* V A - 1 fc^0
(25)
6
sin 6
c(a) =
REMARK have:
1 + cos 0 , - ,
a
7F^ °--¥3r£a*
(26)
The case 7 = 0 leads to r) = 0 or n = —2«0; in both cases we U ^ a , - | i > = ±£ia 0 |0 > + e 2 ^ ( a » ' + c ) lz' >
The operators U(7 = 0(a)) are in this class, however they play a significant role in Grover's algorithm because they may be used to change the sign of a component leaving the others u inaltered (flip). If we are interested in unitaries which modify the component a 0 of a, we must look for solutions with 7 ^ 0 . C O R O L L A R Y 2 If in (21) and (22) we choose: £ e
l 4 — £3 = £5 = 1
e2 = - l N-2 ft N 2 7o= 7V then the corresponding operator U is Grover's unitary (see section 4). P R O O F It is known that Grover's unitary is characterized by (see section 4): TV — 2 2 do H-> — « 0 + "77 Y, Uk = : £1 K + 'K 0 )] (27) 7V
iV
fc^O
a,- H-> - a ; + — I - a 0 + X] a* 1 = : £2 [a* + c ( a )l
(28)
7V
\ Mo / On the other hand, from equations (25) and (26) we have: £i [a0 + f](a)] = £i£ 4 I P0a0 + £s7o 5Z a * I
(29)
e 2 [a; + c(a)] = £2 I a; + 7oao
N
V
1
1 ] «fc ) Mo /
(30)
with 70 given by (5). Comparing this with (27) and (28) we see that the condition for equality is: £l£4P0
N Now let us choose Si£4 — 1 and (30 = —^ N ' then: 1 _
7o = £5
1.
N
(Af-2) 2 N2 _
,
JV-1
9 z
^
that leads to e s = 1. Therefore, if e 2 = — 1, the coefficient of the third term in (30) becomes: -£2
1 + e3/30 N-l
1+ £ 3 ^ TV — 1
N + e3N - 2e 3 iV(JV-l)
that gives the correct parameter ^ if and only if £3 = 1.
2
Canonical form and reflections
T H E O R E M 1 Any unitary operator U(/3 0 , £) with real coefficients in the basis (\i >) and satisfying the conditions of Theorem (1.1), can be written in the form: U ( / ? o , e ) : = e i £ 4 | 0 > Ifo
+ .g|.>( has the form: 1 \u >-
V2
r
/ l + £3/?0
/l+£3/3o\
....
£5^1 - £3^o, ^/^rr' • • •' V ^v^r/
(34)
R E M A R K Notice that unitary operator (32) simply realizes the reflection of the \u >-component of any vector with respect to the \u >-axis. P R O O F The identity (31) follows immediately from (2), (21) and (22). The operator U(/J 0 ,£) of the equation (31) can be rapresented in the following way: l-y _ Q2
U(/?0,e):=£2l->
+ £2 Yl I* >
(£21 - £i£4/30) < 0| - e1e4e3e5^7==^ V < k\ vW - 1 ^ 0
T ^A ^ -£s y/W
1 + £3/?0
r of the form (35), the right end side of (32) is equal to: £ 2 1 - { |0 > - (
+
£2 (1 - £ 3 $ ) ) < 0 | -
£2£5
\ZWg £=
sin 2 ' v ^ V ^ T ' ' " '
(39)
y/N~^J
which, up to a phase, is the most general form of a vector in R components equal. REMARK
with N — 1
A matrix rapresentation of the operator U(/?o, e) in the basis
(\i > ) , w i t h e 2 = £16463 is: 1 + £3$) - 1 ,/l_/3
U(A,,£) = £2
2
y/l-%
£5 Vw-i
£5
£h
l+E 3 /?0 N-l
-
1+esPo N-l
£ 5 - /N-l
l+e 3 A> N-l
1 -
£5-
l+e3/% N-l
-
1 " £3/?o £ 2 ! - £2
l+e3Po N-l
1+esPo N-l
l+e 3 ,3n N-l
-£5-
-1
-Pl \ m^i
^PI
1+E3A) JV-l
ts
l+B3pQ N-l
n-Pl
l+=3/3o
ts
l+e 3 ft) N-l
- £ 5 - 'N-l
-£5-
1 + E3A) N-l
l+e3A) N-l
- V7T0"Pi
c&
^/iv=T
I
/
10
/ i - cos e = e 2 l - e2
sin 8 /N-l
1+cos 8 N-l
1+cos fl N-l
/JV-1
1+cos ( N-l
1+cos ( N-l
sin2 I sin f c o s |
= £ 2 1 - £22
\
sin 8 VN-1
svnj^cos^
\/]V-T
™*2
e
sin ^cos-= VN-1 8 cos^
VT^l
N-l
N-l
JN^l
JV-l
N-l
= £2(1
2|u > < u]
(40)
with \u > given by (35) or (39).
3
Optimal Choice of the Parameters
In this section we study the following generalization of Grover's problem: given a fixed vector \a >= Yliai\i >•, w e l°°k for a unitary operator U — XJ(/30,e) of the form discussed in sections (1) and (2), which increases the probability of the 0-th component of \a >, i.e. such that: \a0\ < |(U(#,,e)|a > ) 0 | := |< 0|U(#,,e)|a > |
(41)
D E F I N I T I O N 1 A unitary operator U(/? 0 ,e) of the form discussed in sections (1) and (2) is an optimal amplifier for the 0-th component of \a > if condition (41) is satisfied and: |(U(A,£)|«>)o|)0|
(42)
V/?0 e [0,1]; V£ := ( £ l , . . . , e 5 ) € {±1} 5 . If moreover: |(U(0 o ,e)|a > ) 0 | = 1 then we speek of an absolute optimal amplifier.
(43)
11 T H E O R E M 2 Given a unit vector of the form: (44)
\aa > : = a0]0 > + 6 ^ \i > with a0 ^ 0 and a20 + (N-l)b2
= l;
(45)
an absolute optimal amplifier exists and is defined by: flo — ±a 0 . P R O O F From equations (21), (22), (25) and (26) we have:
U|a G >:=U L o |0> +bJ2\i >) = £l£4
(30a0 + e3e5y/{N
+£2 b + e5
f1^ y/N~^T
- 1)(1 - ffi)b |0>
a0 - ( 1 + £3/30) b
= £i£4£3 (cos 9aQ + y/N — Isen 9 b) 10 > +£ 2 I v
'
+
El*>=
,
a0 — cos 9 b I V" \i >
Vv/iV-1
)f£
The amplitude of |0 > is extremal if: -jr-T. [COS 9a0 + y/N — lsen 9 b) = —sen 6 a0 + V^V — lcos 9 b = 0 and this is satisfied by a 0 such that: tg 6 = y/N - 1 — a0
(46)
that gives: £6
£3/^0 — cos 9 —
2
£6a0
\ / l + tg 9 and £5
v/i3^
rin 9 =
where £6,£7 € {±1} and we used (45).
e7tg 9
= = e7y/N - 16
From (46) we have e 6 = e 7 .
12
Therefore we obtain: a0 — i > £1£4e3e6 f sen 0 + cos2 0) — exe^e-^e^ and / s e n #cos # b \—> e2£6 — ,
_ _ sen 0cos 0\ ,
= 0
Thus the extremal amplitudes correspond to probability 1 and this completes the proof. R E M A R K The absolute optimality of the previous Theorem (3.2) refers to the case when all the components ctk(k ^ 0) are equal. However, for a general vector, an optimal amplifier will not be absolutely optimal. This fact will be apparent from the following theorem. T H E O R E M 3 An optimal amplifier for a generic vector a of the form:
(47)
|a>:=E«ili> 3=0
with: £"i=l
(48)
3
exists if Y^,kitoak ¥" 0 and it is given by an operator (31) with the following choice: tg 0=
^y°°*.
P R O O F From equations (21), (22), (25) and (26) we have: N
\J\a >:= U V V - | j >= £i£ 4 3=0
/?0a0 + £3£5 /.. \
\A - /% 'VN^Ta°
*o\ / £!£4£3
\A~~ ft2 \
(
COS 0 a o +
° £ afc
V « - l
t #
1 + £3/?o N-l ^ S€-Yl 0 —7=
= Ea*)l°> 1
*/o
/
1° > /
13
r-^ /
sen 0
1 + cos JV-
r^ak) 1
Mo
and the amplitude of |0 > is extremal for: sen 06 ^ sen
d I J-e[cosOa0
+
\
I >
/ cos 0
^
7W
then for a 0 such that:
this gives: £e
£3/30 = cos 6
£ 6 a 0 y/JN^l
£e V
o^(JV-l)
£7tg 9 viV — I70 = sen t> = yjl + tg2 6
£7 Efc^o Q-k ^(7V-1) + (EM0^)
with £6, £7 € ± 1 and £6£7 — 1, i.e. £6 = £7. This gives: a0 H-> e 1 e4e 3 e 6 \al(N - 1) + (Ei#o«fc)2] V/AT^I^AT
eieAe3e&^Jal{N
_ i)
+
(E^Q
flfc)2
- 1) + (Efc^o a i)
y/W^l
k^O k (s + a
£l£4£3£6 1\ a §
N-l
(49)
Finally:
a; h-> £ 2
a; +
£e
%^
r:r
l+£6
zJft^O ak
I ^ ( 7 V - l ) + (E^0a4)
= e2 a, and this completes the proof.
2 a0
^(JV-iJ+CE^o"*)2
N-l
(50)
14
R E M A R K Obviously (49) and (50) shall be as in Theorem (3.2) if the vector \a > is of the form (44). R E M A R K The action of the optimal amplifier found in Theorem (3.3) can be described in the following way: "For every \a >: 1. we subtract from every a;, i ^ 0, the average of all the components different from the 0-th one: a; — i > a,- — N - 1 2. Then for the 0's component we have of course:
'-Sl*-^)'-
a0
I-XX-^-
1
(E/t^O ak) ) -' (N-IY ,
+
(Ei^O aij (E/fc^O ak) _ = JV-1
\T,kjtoak)
(where in the last step we used (48)) as in the (49)".
4
Grover's algorithm
Grover, in [1], considers the following problem (cf also [2]): P R O B L E M : Given a (quantum) system with a state space H of dimension N = T. Let { 0 , 1 } ^ = {S0,S2,-..,SN-i} =: S be the set of states represented as n q-bit string £ 7i. Let be given a function: C:S^
{0,1}
15
with the following property: there is only one state, say Sv, such that C(SV) — 1, while C(S) — 0 V5 7^ Sv. Construct a quantum computer algorithm which is able to find the unknown Sv state with high (say > .5) probability. It is always possible to rename the states so that {Si,..., SV} = { 0 , . . . , N1} and Sv = 0. In these notations let be given a vector:
\a >:— E a «N > i
The first step in Grover's algorithm is to construct an operator Z that flips the 0-component. In our notations: Z := 1 - 2 | 0 > < 0 | Grover then defines: \a >: = Z\a > = — ao\0 > + ^ J a t |i > and chooses the unit vector in formula (32) as follows: \v >:= —i= VIA: > = —^=|1,...,1 >
(51)
This gives:
=^E + g^>]=^(-o + E^) Then, denoting P := \v > < v\, Grover introduces the unitary operator D|a >:= —1 + 2 P , whose action on \a > is given by: D|a >:= (-1 + 2P) \a >= -\a > +2 < v\a > \v >= -\a > +—=
|o>+E Then:
' +N
-a0 + E a t
l^\-a0+J2ak
TV — 2 2 a do !-> —TJ— ao + T7 k = £ i K + »?(a)] ,V iV i5Z ^O
\v >
\l
>
kjtO
(52)
16
a{
2
H->
-a{ + — j - a 0 + ] T a M = e 2 [ai + c («)
(53)
TV
If < u|)(l - 2|0 > < 0|) = = - 1 + 2\v >< v\ + 2|0 >
in the following way: O l + 2|u > < H + 210 > < 01 N-l a
° + Jf g
1
\v > < 0 | ) | a > = N-l
a
i - ^ « o , U + jj g
a,- - - a o TV
i=i,...,yv-i
-^)* + IS*" (l"°-- + ^£°')j=1 „_,
(54)
Comparing (54) with (49)and (50) we conclude that, within the class U(/3 0 , e), Grover's unitary is not optimal, not only for a generic initial vector \a >, but even for the initial vector \aG > used by Grover himself and given by (45).
17
5
A One Step Solution for Grover's Problem
In the notations of section (4) let us write \j > for the state Sj so that, in particular, \SV >= \v >, and suppose moreover that j = 0 , 1 , . . . , N — 1. In Grover's algorithm one uses the unitary operator: vb'>=(-i)c(5j)|j> which is a self adjoint involution, i.e. V = V* and V = 1 We will consider the unitary operator: f \j >
V | j > = { |0> { \j >
rfJ^Oor
ifC{Sj)
C(S3) = 0
=l
if j = 0 and C(SV) = 1
2
which is also an involution (V = 1). Notice that \\j >— J2k ujk\k > and Ujk = &jk if j ^ v, 0; uvo = 1, uvf. — 0 for k ^ v, UQV = 1, Uok — 0 for k ^ v. Thus V is a local operator in the sense of Grover [1], section (5). The action of V can be compactly described by:
v
l i > = (1 - Kc(s0))6o,c{Sj)\j
> + +(1 l
+ z
{Sk)
9J2{l-(-lf
}\k>
6liC(s0))6j,o+
(55)
k=i
from which it is clear that the physical action of V is realized by parallel computation of the values of C, exactly as in Grover's algorithm. T H E O R E M In Theorem (3.2) let us choose the initial vector \aG > in (45) so that a0 — b — -j= (i.e. we choose Grover's initial vector) and let us denote U the corresponding absolute uptimal unitary operator. Define: UOPT
:= V*UV
then U O P T is a n absolute optimal amplifier for the component \SV > of \aa >. P R O O F Clear from theorem (3.2) and the definition of U.
18
6
APPENDIX A: Direct proof that U verifies the Isometricity Condition
Let us now verify that the set of conditions (4), . . . , (8) are also sufficient conditions. To this goal we check if the isometricity condition (11) is satisfied by the operator U, given by (3) if the parameters satisfy (4), . . . , (8). Then, replacing (21) and (22), which are equivalent to (4), . . . , (8), into (10) and the (12), we find: r,(a)2 = ( _ l + £ 4 / ? 0 ) 2 a 2 + i ^ ( £ a J A - -1 \kjto /
+
2£4e3£5(Zo(_1+£4/?o)^L^^a)t V iV - 1 klt0 _ ai / . . 1 Yl a *
IY
2a0rj(a) - 2 ( - l + e4/?o)ao + 2e4e3£5a0
r,(ay + 2aoV(a) = (-l + f320)al + j
r
^ 1( 2g^ >ak\) + + 2e 2s33ee5a50af30f3 0^-0^=J^ 1^0
(iV-l)c(a)2 = (l-/%)«3+^l 3 ffi
/
ga
V JV -
1 ^
k
0
( E«*J - 2 £ 5 a 0 ^ = = ( l + ^ o ) E ^ \kjto J yjv i ^ 2
o
n
r
2c(a) MO ^,afc
>/l ' -^ /^o ^ l + £ 3 / 3 o Y- ' V ' V 0£+£3^o = 2 e 5 *y/W=\& a 0 ^ 3 Y ^ ;aka * " 2 yv - 1 lW^ o° * N-l 0
Therefore the isometricity condition (11) is equivalent to: 2
!
,( =
-. = ( - i + a w - 4 ^ W + ^ ^ + - « g < , +
19
0
V
1
-
^ ^
l 0
lt£3Po
'
r
or equivalently: 2
2e3£e3 /.. , F=I> ^o-iK + ^ r Y ( Z^ E k\^ J ++2 5qoPo £5«o/?o^ a
k?o
J
V^ - 1 ^0
2
-(-1 + AK +
W^2
(EH +
A _ Q2 +2e 5 ao^7===^(l + e3A) - 1) J2 a * v Jv - 1 ^0 which is an identity for any choice of e 3 , e 4 , e 5 and this ends the proof.
7
Bibliography
[1] Lov K. Grover: Quantum Mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79, 325-328 [2] Boyer M, Brassard G, Hoyer P and Tapp A: Tight bounds on quantum searching (preprint quant-ph/9605034) [3] Luigi Accardi, Ruben Sabbadini: On the Ohya-Masuda quantum SAT algorithm Volterra preprint 2000, [4] M. Ohya and N. Masuda, NP problem in Quantum Algorithm, quantph/9809075. [5] M. Ohya, Mathematical Foundation of Quantum Computer, Publ. Company, 1998.
Maruzen
[6] A. Yu. Kitaev: Quantum measurement and the abelian stabilizer problem, quant-ph/9511026, 20-11 (1995) [7] M. Ohya, N. Watanabe: On Mathematical treatment of Fredkin-ToffoliMilburn gate, Physica D, 120 (1998) 206-213
20
[8] I.V. Volovich, Quantum Computers and Neural Networks, Invited talk at the International Conference on Quantum Information held at Meijo University, 4-8 Nov. 1997, Proc. of the Conference. [9] I.V. Volovich, Models of quantum computers and decoherence problem, preprint Vito Volterra N358, 1999. [10] I.V. Volovich, Mathematical Models of Quantum Computers and quantum decoherence problem, in Volume dedicated to V.A. Sadovnichij. Moscow State University, 1999, to be published. [11] I.V. Volovich, Quantum Kolmogorov machine, Invited talk at the International Conference on Quantum Information held at Meijo University, 1-6 March 1999, Proc. of the Conference. [12] I.V. Volovich, "Atomic Quantum Computer", quant-ph/9911062; Volterra preprint N. 403, 1999, Universita degli Studi di Roma "Tor Vergata". [13] I.V. Volovich: Models of quantum computers and decoherence problem. Volterra preprint N. 358, 1999, Universita degli Studi di Roma "Tor Vergata". [14] M. Ohya, I.V. Volovich: Quantum computing, NP-complete problems and chaotic dynamics. Volterra preprint N. 426, 2000, Universita degli Studi di Roma "Tor Vergata". [15] Luigi Accardi, Masanori Ohya, Ruben Sabbadini: in preparation
Quantum Information III, pp. 21-33 Eds. T. Hida and K. Saito © 2001 World Scientific Publishing Company
Characterization of Product Measures by an Integrability Condition Nobuhiro ASAI International Institute for Advanced Studies Kizu, Kyoto 619-0225, Japan. asaiQiias.or.jp
1
Introduction
Let (£',n) be the real Gaussian space, where £' is the space of tempered distributions and ft be the standard Gaussian measure on £'. In the recent papers [4, 7, 8], Asai, Kubo and Kuo (AKK for short) have shown that in order to construct the Gel'fand triple [£]„ C L 2 (£',ti) C [£]£ associated with a growth function u € C+?i/2> essential conditions on u are (U0)(U2)(U3) stated in Section 2.2. Legendre transform and dual Legendre transform (Section 2.1) play important roles to get this result. We note that Gannoun et al. [11] have obtained similar results independently. Some relationships with [11] are discussed in Section 2.2. In addition, the intrinsic topology for [£]„ has been given and the characterization theorem for positive Radon measures on £' has also been proved by considering an integrability condition [5, 8]. Now it is natural to ask whether "positivity" of white noise operators can be discussed in some sense and characterized. To answer this question, we consider the Gel'fand triple over the Complex Gaussian space (£'c,fic), i.e. £'c = £' + i£' equipped with the product measure fic = \i' x y! where fi' is the Gaussian measure on £' with variance 1/2 (Section 2.2). Following AKK's Legendre transform technique, we have W „ u „ 2 C L2(£'c,ftc) C [W]£ l U 2 for functions «x,«2 € C + ] i/2 satisfying (U0)(U2)(U3). Several examples for « i , u 2 are given in Section 2.3. We remark that Ouerdiane [28] studied a special case « i ( r 2 ) = u 2 ( r 2 ) = exp(fc - 1 r*), where 1 < k < 2. In Section 3, the characterization theorem for measures can be extended to the case of positive product Radon measures on £' x £'. In addition, the notion of pseudo-positive operators is naturally introduced via kernel theorem and characterized by an integrability condition. Lemma 3.2 plays crucial roles in Section 3.
2
White Noise Functions
2.1
Legendre Transform and Dual Legendre Transform
In this section we introduce the Legendre transform and dual Legendre transform which will be used for the constructions of the Gel'fand triples over the real and complex Gaussian spaces. First, let us define two kinds of convex functions. A positive function / on [0, oo) is called (1) (log, exp)-convex if the function l o g / ( e x ) is convex on R;
21
22 (2) (log, xk)-convex
if the function l o g / ( x * ) is convex on [0,oo). Here k > 0.
Let C+^iog denote the collection of all positive continuous functions u on [0, oo) satisfying the condition: log u(r) hm —p—^—^ = oo. r-+oo
log r
The Legendre transform lv of u 6 C^iog is defined to be the function
eu{t) = inix^-, v
<e[o,oo).
r>o r (
Let C + j l / 2 denote the collection of all positive continuous functions u on [0, oo) satisfying the condition: hm r-+oo The o u(s)
re[0,oo).
Note that C+ii/2 C C+,i og - Assume that u G C+,iog and lim n _ > 0 0 ^„(n) 1 /" = 0. We define the L-function Cu of u by oo
For discussions in the rest of the paper, we will need the following facts in [7, 8]. See also [1]. Fact 2 . 1 . (1) Let u G C+jog be (log, exp)-convex. also (log, exp)-convex and for any a > 1, ect
£ „ ( r ) < -, u(ar), logo
Then its L-function
Vr > 0.
(&) Let u g C+,log be increasing and (log, xk)-convex. constant C, independent of k, such that u(r) < CX„(2*r)>
Cu is
Then there exists a
Vr > 0.
f#j £et u 6 C+,iog 4e increasing and (log, xk)-convex. have
(2.2) Then for any a > 1, we
£u(r)