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(|0>|0> + |1>|1»,
|*-> = - = (|0>|0>-|1>|1». •v/2
(2)
This is the so-called Bell basis. It is important to notice that here we can still encode two bits of information, that is we have four different possibilities, but now this encoding is done in such a way that none of the bits carries any welldefined information on its own. All information is encoded into relational properties of the two qubits. It thus follows immediately that in order to read out the information one has to have access to both qubits. The corresponding measurement is called a Bell-state measurement. This is to be compared with the classical case where access to one qubit is simply enough to determine the answer to one yes/no question. In contrast, in the case of the maximally entangled basis access to an individual qubit does not provide any information. 3. Quantum communication and dense coding Whenever, say, two parties A (Alice) and B (Bob) wish to communicate with each other they have to agree first on a coding procedure, that is they have to agree which symbol means what. In classical coding the situation is very simple. Restricting ourselves to binary information, that is to bits, we need some information carrier which has two states. A famous historic example from the American revolution was when Paul Revere informed the Revolutionaries about the path taken by the Royals by displaying one or two lamps in the steeple of Old North Church in Boston. In quantum physics again we can have information encoding in a novel way using entangled states and thus encode information into joint properties of elementary systems. Then the elementary systems themselves do not carry any information. A first elementary case where this is clearly demonstrated is quantum dense coding. The maximally entangled Bell basis of eq. (2) has a very important and interesting property which was exploited by Bennett and Wiesner [10] in their proposal for quantum Physica Saipta T76
dense coding. This is the property that in order to switch from any one of the four Bell states to all other four it is sufficient to manipulate only one of the two qubits while in the classical case one has to manipulate both. Thus, the sender Bob (Fig. 1) can actually encode two bits of information into the whole entangled system by just manipulating one of the two qubits. Let us, for example, assume that we start from the state | lP+> then we can obtain | f~> by just introducing a phase shift of n onto, say, the second qubit, | -
|f
S
>=-
:(|a>|ft>-|b>|a», ]a>|fc> + |ft>|a»,
(3)
where | TA} and | Wsy are anti-symmetric or symmetric respectively. Because of the requirement of symmetry the total two-photon states are !^+>|fs>, |f">|fx>. |0
+
>|fs>,
l^>l"F s >-
(4)
We note that only the state anti-symmetric in external variables is also anti-symmetric in internal variables. It is this state which also emerges from the beam splitter in an external anti-symmetric state. This can easily be found by assuming firstly that the beam splitter does not influence the internal state and secondly that at the beam splitter we have a phase shift of jc/2 upon reflection. Actually it can easily be seen [13] that the spatially anti-symmetric state is an eigenstate to any beam splitter operator. In contrast, in all three cases of the symmetric external state | fs> the two photons emerge together in one of the two outputs of the beam splitter. This behaviour of symmetric states can readily be demonstrated even using photons in an unentangled internal state [14]. Figure 3 and 4 show the consequent experimental behaviour [15] for the entangled internal state | Y+y and that for the internally antisymmetric state | Y~). It is therefore evident that the state \V~y can clearly be discriminated from all the other states. It is the only one of the four Bell states which leads to coincidences between
205
10000 7500 5000 2500
g '3
u
Flight Time Difference i (ps) Fig. 3. Coincidence counts behind a beam splitter as a function of the difference of arrival time of two photons in an internally symmetric entangled state. For large time differences the two photons are distinguishable and therefore they behave independently and a certain coincidence rate equal to the classical rate is observed. For zero time difference there are no coincidences because both photons always leave the beam splitter together in the same output port.
detectors placed on each side after a beam splitter. How can we then identify the other three states? It turns out that distinction between 11?+> on the one hand and | $ + > and | 0} on the other hand can be based on the fact that only in | W+y the two photons have different polarization while in the other two they have the same polarization. Thus performing polarization measurements and observing the photons on the same side of the beam splitter one can determine whether they are in the state \W+) or in one of the states | 0 + } and | iy s >. l* _ >l^>.
(5)
because of the antisymmetry requirement of the total state. For fermions therefore only one of the states is spatially symmetric, the other three are spatially anti-symmetric. It will thus be in only one of the cases, namely for | I P - ) that both fermions will emerge together from the beam splitter. In all other three cases they will emerge from different sides [16]. Yet, remarkably, it is again this state which can immediately be distinguished from the other three because of its distinct symmetry properties. 5. Entanglement swapping and the nature of quantum information Entanglement used to be considered as a consequence of the fact that the entangled particles interacted in their past or that they came from a common source. That this is too restricted a view is witnessed by the concept of entanglement swapping [17]. In the simplest case of entanglement swapping we take two entangled pairs (Fig. 5) and subject two particles, one from each source, to a Bell-state measurement. Then the other two particles which have never interacted in the past and also did not come from a common source are projected into an entangled state. This experiment, whose realization is presently proceeding in Innsbruck [31], has some deep implications for the meaning of information in quantum physics. As mentioned above, entanglement between two quantum systems implies that at least some information is stored only in joint or relational properties of the two systems. In the case of maximal entanglement there is actually no information stored in any individual particle, all information is carried by all the particles jointly. This fact in itself poses already a challenge for a naive realist position. How can it be that particles have well-defined joint properties - for example when we know that their polarizations in a certain basis are equal without carrying any properties by themselves? For a Bell state each of the systems is completely unpolarized on its
Bell-state measurement
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Fig. 5. In entanglement swapping two Bell sources I and II emit one pair of entangled qubits each. One then performs a Bell-state measurement on a pair consisting of one qubit from each source. This projects the two outermost qubits which have never interacted onto an entangled state. Physica Scripta T76
own? In the case of entanglement swapping this interpretive problem is even more striking. Let us, for simplicity of discussion, just consider the case where we have sources that produce our two qubits in the anti-symmetric state | IP - ). This state has the unique feature that it is anti-symmetric in any basis. Thus, in terms of its information content, the statement is that we know the two qubits are different whatever basis we choose1. We thus know simply by the choice of preparation that in each of the two entangled pairs to be used in entanglement swapping the two qubits are completely different. We now have to discuss the information content of Bellstate analysis. At first we note that, whichever states we would produce at the sources, a fair Bell-state analyser will return any of the four possible answers with equal probability of 25%. That is, the action of the Bell-state analyser is such that it projects the two photons onto an entangled state and, since in our case the two qubits are themselves members of maximally entangled pairs and therefore carry no information, this has to happen with equal probability for all four Bell states measured. In fact, the Bell-state measurement does not reveal any information about any of the qubits emitted by one of the two sources nor any joint information about each source. Yet, what we gain is joint (or relative) information about the two sources. Suppose, specifically, that in a certain experimental run we obtain the result | IP - ) for the Bell-state measurement. We then know that qubits 2 and 3 have been projected by the measurement onto a state which is characterized by the fact that these two qubits are different in whatever basis. Interestingly, and again most remarkably, this statement is even true as none of the two qubits themselves are yet welldefined in any basis and have no properties by themselves. Now we are in a position to complete our chain of reasoning. By the properties of source 1 we know that qubit 1 and qubit 2 are different. By the result of the specific Bellstate measurement we know that qubit 2 and 3 are different and, finally, from the property of source 2 we know that qubit 3 and 4 are different. Therefore, since our qubits are defined in a Hilbert-space of dimension 2 only, we conclude that qubits 1 and 4 also have to be different in any basis. Therefore they emerge in the anti-symmetric state ]!P - ). Analogous reasoning can be built up for the other three possible Bell-state measurement results. This analysis supports the interpretation that quantum mechanics is just a formulation of what can be said about quantum systems very much in the spirit of Bohr [18]. The question as to what "really" the properties of these quantum systems are is empty and devoid of any meaning. We could go even one step further realising that in quantum physics information has a novel quality which goes beyond its nature in classical physics. In classical physics it is natural to assume that the objects we observe
1
We remark that for the three symmetric states we can make analogous statements but the situation is slightly more complicated. It turns out in the end that, since the Hilbert-space of the four Bell-states is fourdimensional, we can encode two independent bits of information into these four states. Therefore the four states can be characterised by yes/no answers to two distinct questions. These two questions are questions about the identity of the two qubits in two different bases conjugate to each other.
Quantum Entanglements
207
the case of entanglement swapping, Alice will obtain each one of the four possible answers with equal probability, that is her original qubit and her qubit from the ancillary pair will be projected onto any one of the four Bell-states each with probability 25%. We note again that this measurement does not reveal any information, neither about the properties of the original qubit nor about the properties of the ancillary pair. So Alice obtains one of four possible results, •«ip+'^ "ip->^ "(p+"t o r " $ " . She then broadcasts this information, that is two classical bits, such that Bob can receive them. By now Bob is in posssession of a specific state as a consequence of Alice's Bell-state measurement. Performing one of four unitary transformations depending on Alice's specific result Bob can transform his particle into the original qubit. We also note that the original qubit disappears during the Bell-state measurement, it looses its identity, and thus Bob's qubit is not a copy but really a teleported 6. Quantum teleportation and the non-locality of information reappearance of the original. A most remarkable application of the concept of entangleA remarkable property of the scheme is that the four ment is quantum teleportation. Consider first the problem. unitary operations Bob has to perform are completely unreSuppose that Alice has an object which Bob, who could be lated to the state Alice wants to teleport. This is because anywhere, might need at a certain time. In classical physics Alice's Bell-state measurement does not convey any inforwhat she can do is perform many precise measurements on mation about the original state. the object and send the information to Bob who then can The experiment can most easily be understood again on reconstitute the object. Within classical physics the accuracy the basis of the fact that the meaning of entanglement is that of this is only limited by the precision with which Alice can it represents relational statements between quantum measure and by the technical abilities of Bob. Yet, we know systems. So, in the case of teleportation we have to use that in the end any measurement will run into the limi- entanglement twice, firstly when the ancillary pair is productations imposed by quantum mechanics. It is evident that ed and secondly in the Bell-state measurement. It thus turns no measurement whatsoever performed by Alice can reveal out that the entanglement of the ancillary pair implies a the full quantum state of the object. We therefore ask which well-defined relation between these two qubits without strategy can Alice pursue in order for Bob to obtain the defining their individual properties, and the Bell-state meaobject in its full quantum state when he needs it. surement implies a well-defined relation between the qubit A strategy proposed by Bennett et al. [19] uses exactly the to be teleported and one of the ancillaries. Therefore, the information-theoretic features of entanglement mentioned second ancillary, by the same chain of reasoning as above in above. According to that protocol, Alice and Bob have to our discussion of entanglement swapping, is in a wellshare in advance an entangled pair of quantum states (Fig. defined relationship to the original qubits to be teleported. 6). Let us consider for simplicity that the object is simply a It is again this remarkable feature of information in two-state system, a qubit. Then Alice and Bob share from quantum mechanics that we have a closed chain of logical the beginning an ancillary entangled pair which for conve- reasoning about relational properties without any statement nience we again consider to be in the state j !P~). Subse- on the properties of the individual systems. quently Alice performs a Bell-state measurement on her The major problem in the experimental verification of qubit and one of the two ancillaries. As discussed above, for quantum teleportation [20] is the Bell-state measurement of two independently created quantum systems. This means BOB that these two qubits have to be measured such that their identity is lost, that it is not possible to infer which detection ^yljeir^ Unitary event refers to which source. It turns out that in the experi(Source) ' Operation ment it is rather tricky to achieve this situation. As can be seen in [21], it involves an elaborate application of a quantum erasure technique. exist prior to and independent of any measurement and the information we gain by measurement is an entity of secondary quality while the objects are primary. In contrast it is suggestive that in quantum physics the situation is that information can be a primary quality and the objects with their properties have a secondary quality, they can be constructed in retrospect after the information has been gained in a measurement. This is consistent with Bohr's Caveat that we should be very cautious about making mental pictures of quantum systems. Such pictures are only allowed in the context of a specific experimental setup. In fact, most of the misunderstandings and misinterpretations of quantum mechanics result from assuming that quantum objects have properties prior to and independent of the specific apparatus chosen to observe them.
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7. Multi-qubit entanglement in quantum computation Bell-state measurement
The most complex application of quantum entanglement discussed presently is quantum computation [22]. Quantum computation heavily uses superposition of multi-qubit ALICE systems, i.e. entanglement. Therefore it is natural that Fig. 6. In quantum teleportation Alice performs a joint Bell-state measure-among the most basic paradigms in the field are the proment on her initial particle and on one of an entangled ancillary pair from duction and analysis both of Bell states and of so-called the Bell source. This then projects the other particle of the ancillary into a GHZ states [23], i.e. three-qubit entanglements. state which Bob can bring into exactly the original state of the initial partiFrom the point of view of the novel possibilities introcle using one of the four unitary transformations according to Alice's Bellduced by quantum computation it is important to realize state measurement result. Physica Scripta T76
208
A.
Zeilinger
that superposition of multi-qubit states makes massive quantum parallelism possible [24]. This is that fact that a quantum computer can produce the result of various different inputs at the same time if the input state is a superposition of the individual informations. Consider, for example, that using standard binary encoding the most simple G H Z state can be interpreted as a superposition of two numbers
4=(|000> + |1H> = 4=(|"0"> + l"7"» V2 V2
(fi)
that is, it represents a superposition of the numbers "0" and "7". Taking this as the input to a linear quantum computer it will produce in the output a superposition of the results corresponding to the two inputs. It is well known that a most important practical problem a future quantum computer faces is decoherence due to coupling to the environment. Somewhat analogous to classical computers decoherence can be viewed as a quantum analogue of classical noise and it can be overcome using redundant information. Therefore, quantum error correction codes [25] utilize the possibility of encoding n qubit into the states of a Hilbert-space of higher dimension JV. One then exploits the additional degrees of freedom to eliminate the quantum noise. This, naturally, implies entangled qubits in a Hilbert-space of high dimension. While the field of quantum computation is signified presently by development of a vast number of different codes and theoretical approaches, the status of experiments is to verify the individual elementary steps including two-qubit quantum gates. It is presently completely open whether future development will utilize multi-photon states as discussed in this paper, quantum encoding into individual atoms [26] or individual ions [27] in trap configurations or bulk nuclear magnetic resonance techniques [28], Yet, the fundamental individual steps in all these techniques are all the same from a quantum information science point of view.
8. Concluding comments There are several further applications of entanglement in information science. One of them is the use of E P R pairs in quantum cryptography as proposed by Ekert [29]. There one actually uses a test of Bell's inequality to find out whether or not an eavesdropper has gained any access to the qubits sent. Another concept utilizing entanglement is quantum computation in distributed networks [30]. This concept presently is in the stage of ideas and has not yet found experimental verification. It would imply extensive use of quantum teleportation, of Bell-state measurements, of entanglement swapping, and of multi-particle entanglements. Taken together, all the examples given above demonstrate already now that quantum entanglement is one of the most important novel concepts in information science. It is furthermore not unreasonable to even expect future applications of entanglement in information technology itself. This is supported by the observation that the present development of information technology towards further and further miniturization will reach the single quantum limit sometime in the first decades of the next century. Once that Physica Scripta 776
happens information technology will by necessity be quantum. The radically new features of quantum entanglement let us expect that this will not just be a limitation but that it will open up completely novel possibilities for communication and computation. The intricate interpretive questions raised by the multiple use of entanglement in the examples given here and as discussed above underline once more the deep conceptual problems of quantum reality. It is hard to believe that there is a possibility to escape Niels Bohr's verdict "There is no quantum world, there is only an abstract quantum description".
Acknowledgements The author thanks the Nobel Committee for its hospitality. This work was supported by the Austrian Science Foundation (FWF), project S65/04 and by the US National Science Foundation (NSF), giant number PHY9722614. I also wish to acknowledge the collaboration with all members of my group at the University of Innsbruck.
References 1. Bohr, N, "Discussions with Einstein on Epistemological Problems in Atomic Physics" in Albert Einstein: Philosopher-Scientist" (Edited by P. A. Schilpp) (The Library of Living Philosophers, Evanston 1949), p. 200. 2. Einstein, A., Podolsky, B., and Rosen, N, Phys. Rev. 47, 777 (1935). 3. Bohr N, Phys. Rev. 48, 696 (1935). 4. Bohm, D., "Quantum Theory" (Prentice-Hall, Englewood Cliffs, NJ, 1951) p. 614. 5. Freedman, S. J. and Clauser, J. S, Phys. Rev. Lett. 28, 938-941 (1972). 6. Aspect, A, Grangier, P. and Roger, G., Phys. Rev. Lett. 47, 460 (1981); 49, 91 (1982); Aspect, A., Dalibard, J. and Roger, G., ibid. 49, 1804 (1982). 7. Kwiat, P. G., et al. Phys. Rev. Lett. 75, 4337 (1995). 8. Pearle, P. M., Phys. Rev. D2, 1418 (1970); Clauser, J. F. and Shimony, A., Rep. Prog. Phys. 41, 1881 (1978). 9. Greenberger, D., Home, M. A. and Zeilinger, A., "Bell's Theorem, Quantum Theory, and Conceptions of the Universe", (Edited by M. Kafatos) (Kluwer, Dordrecht, 1989) p. 69; Greenberger, D. M., Home, M. A, Shimony, A. and Zeilinger, A., Amer. J. Phys. 58,1131 (1990). 10. Bennett, C. H. and Wiesner, S. J., Phys. Rev. Lett. 69, 2881 (1992). 11. Mattle, K., Weinfurter, H., Kwiat, P. G. and Zeilinger, A., Phys. Rev. Lett. 76, 4656 (1996). 12. Michael Reck, unpublished. 13. Zeilinger, A., Bernstein, H. J. and Home, M. A., J. Modern Opt. 41, 2375 (1994). 14. Hong, C. K, Ou, Z. Y. and Mandel, L., Phys. Rev. Lett. 59, 2044 (1987). 15. Oberparleiter, M., Diploma-thesis, University of Innsbruck, 1997, unpublished. 16. Loudon, R., "Coherence and Quantum Optics VI" (Edited by J. H. Eberly and L. Mandel) (New York: Plenum, 1990), p. 703; Loudon, R., "Disorder and Condensed Matter Physics" (Edited by J. A. Blackman and J. Taguena) (Oxford: Clarendon, 1991), p. 441. 17. Zukowski, M-, Zeilinger, A., Home, M. A. and Ekert, A. K., Phys. Rev. Lett. 71, 4287 (1993). 18. Nagel, B., in "Possible Worlds in Humanities, Arts and Sciences". (Proceedings of Nobel Symposium 65) (Edited by Sture Allen) (Gruyter, Berlin 1989), p. 388. 19. Bennett, C. H. et al, Phys. Rev. Lett. 70, 1895 (1993). 20. Bouwmeester, D. et at., Experimental Quantum Teleportation, Nature 390, 575 (1997). 21. Zukowski, M., Zeilinger, A. and Weinfurter, H., "Fundamental Problems in Quantum Theory" (Edited by D. M. Greenberger, A. Zeilinger) (Annals of the New York Academy of Sciences, 1995) vol. 755, p. 91.
Quantum Entanglements 22. 23.
C. H. Bennett, Phys. Today 48, 24 (1995). Brass, D , Ekert, A , Huelga, S. F., Pan, J.-W. and Zeilinger, A., Phil. Trans. R. Soc. Lond. A (1997) in press. 24. Shor, P. W., in Proc. of the 35th Ann. Symp. on the Foundations of Computer Science, Los Alamitos, CA, (IEEE Press, New York 1994); Ekert, A., Jozsa, R., Rev. Mod. Phys. 68, 733 (1996). 25. Shor, P. W., Phys. Rev. A52, 2493 (1995); Steane, A. M , Proc. Roy. Soc. Lond. A4S2, 2551 (1996). 26. Hagley, E. et al, Phys. Rev. 79, 1, (1997).
209
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Monroe, C. et al., Phys. Rev. Lett. 75, 4714, (1995); Wineland, D., this volume. 28. Gershenfeld, N. A. and Chuang, I. L , Science 275, 350 (1997). 29. Ekert, A. K., Phys. Rev. Lett. 67, 661 (1991). 30. Cirac, I., Zoller, P , Kimble, J. and Mabuchi, H., Phys. Rev. Lett. 78, 3221 (1997); Bose, S., Vedral, V. and Knight, P. L„ Phys. Rev. A57, 822 (1998). 31. Pan, J.-W., Bouwmeester, D., Weinfurter, H. and Zeilinger, A., Phys. Rev. Lett. 80, 3891 (1998).
Physica Scripta T76
Experiment and the foundations of quantum physics Anton Zeilinger* Institut fur Experimentalphysik, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Instead of having to rely on gedanken (thought) experiments, it is possible to base this discussion of the foundations of quantum physics on actually performed experiments because of the enormous experimental progress in recent years. For reasons of space, the author discusses mainly experiments related to the Einstein-Podolsky-Rosen paradox and Bell's theorem, that is, to quantum entanglement. Not only have such fundamental experiments realized many historic proposals, they also helped to sharpen our quantum intuition. This recently led to the development of a new field, quantum information, where quantum teleportation and quantum computation are some of the most fascinating topics. Finally the author ventures into a discussion of future prospects in experiment and theory. [S0034-6861(99)03602-8]
CONTENTS I. The Background II. A Double Slit and One Particle III. A Double Slit and Two Particles IV. Quantum Complementarity V. Einstein-Podolsky-Rosen and Bell's Inequality VI. Quantum Information and Entanglement VII. Final Remarks and Outlook Acknowledgments References
S288 S288 S289 S291 S292 S295 S296 S296 S297
I. THE BACKGROUND
Quantum physics, a child of the early 20th century, is probably the most successful description of nature ever invented by man. The range of phenomena it has been applied to is enormous. It covers phenomena from the elementary-particle level all the way to the physics of the early universe. Many modern technologies would be impossible without quantum physics—witness, for example, that all information technologies are based on a quantum understanding of solids, particularly of semiconductors, or that the operation of lasers is based on a quantum understanding of atomic and molecular phenomena. So, where is the problem? The problem arises when one realizes that quantum physics implies a number of very counterintuitive concepts and notions. This has led, for example, R. P. Feynman to remark, "I think I can safely say that nobody today understands quantum physics," or Roger Penrose (1986) to comment that the theory "makes absolutely no sense." From the beginning, gedanken (thought) experiments were used to discuss fundamental issues in quantum physics. At that time, Heisenberg invented his gedanken gamma-ray microscope to demonstrate the uncertainty principle while Niels Bohr and Albert Einstein in their famous dialogue on epistemological problems in what was then called atomic physics made extensive use of gedanken experiments to make their points.
Now, at the end of the 20th century, the situation has changed dramatically. Real experiments on the foundations of quantum physics abound. This has not only given dramatic support to the early views, it has also helped to sharpen our intuition with respect to quantum phenomena. Most recently, experimentation is already applying some of the fundamental phenomena in completely novel ways. For example, quantum cryptography is a direct application of quantum uncertainty and both quantum teleportation and quantum computation are direct applications of quantum entanglement, the concept underlying quantum nonlocality (Schrodinger, 1935). I will discuss a number of fundamental concepts in quantum physics with direct reference to experiments. For the sake of the consistency of the discussion and because I know them best I will mainly present experiments performed by my group. In view of the limited space available my aim can neither be completeness, nor a historical overview. Rather, I will focus on those issues I consider most fundamental.
II. A DOUBLE SLIT AND ONE PARTICLE
Feynman (1965) has said that the double-slit "has in it the heart of quantum mechanics. In reality, it contains the only mystery." As we shall see, entangled states of two or more particles imply that there are further mysteries (Silverman, 1995). Nevertheless, the two-slit experiment merits our attention, and we show the results of a typical two-slit experiment done with neutrons in Fig. 1 (Zeilinger et al, 1988). The measured distribution of the neutrons has two remarkable features. First, the observed interference pattern showing the expected fringes agrees perfectly well with theoretical prediction (solid line), taking into account all features of the experimental setup. Assuming symmetric illumination the neutron state at the double slit can be symbolized as
\)*Electronic address: [email protected] S288
Reviews of Modern Physics, Vol. 7 1 , No. 2, Centenary 1999
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©1999 The American Physical Society
Anton Zeilinger: Experiment and quantum physics
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FIG. 1. A double-slit diffraction pattern measured with very cold neutrons with a wavelength of 2 nm corresponding to a velocity of 200 ms"1. The two slits were 22 fim and 23 fjucn wide, respectively, separated by a distance of 104 /im. The resulting diffraction angles were only of the order of 10 /trad, hence the observation plane was located 5 m downstream from the double slit in order to resolve the interference pattern. (For experimental details see Zeilinger et al, 1988.) The solid line represents first-principles prediction from quantum mechanics, including all features of the experimental apparatus. For example, the fact that the modulation of the interference pattern was not perfect can fully be understood on the basis that a broad wavelength band had to be used for intensity reasons and the experiment was not operated in the Fraunhofer regime. The interference pattern is then obtained as the superposition of two probability amplitudes. The particle could have arrived at a given observation point f either via slit 1 with probability amplitude a(f) or via slit 2 with probability amplitude b{r). The total probability density to find the particle at point f is then simply given as p(n = \a(P) + b(F)\2.
(2)
This picture suggests that the pattern be interpreted as a wave phenomenon.
S289
Yet, second, we note that the maximum observed intensity is of the order of one neutron every two seconds. This means that, while one neutron is being registered, the next one to be registered usually is still confined to its uranium nucleus inside the nuclear reactor, waiting for nuclear fission to release it to freedom! This feature of very low-intensity interference is shared by all existing neutron interferometer experiments (Rauch and Werner, in press). These pioneering matter-wave experiments led to the realization of a number of very basic experiments in quantum mechanics including the change of the sign of a spinor under a full rotation, the effect of gravity on the phase of a neutron wave, a number of experiments related to quantum complementarity, and many others. Thus the interference pattern is really collected one by one and this suggests the particle nature. Then the famous question can be posed: through which of the two slits did the particle actually pass on its way from source to detector? The well-known answer according to standard quantum physics is that such a question only makes sense when the experiment is such that the path taken can actually be determined for each particle. In other words, the superposition of amplitudes in Eq. (1) is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still "out there." The absence of any such information is the essential criterion for quantum interference to appear. For a parallel discussion, see the accompanying article by Mandel (1999) in this volume. To emphasize this point, let us consider now a gedanken experiment where a second, probe, particle is scattered by the neutron while it passes through the double slit. Then the state will be
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V2 + |passage through slit 6)[Scattered in region
There the subscripts 1 and 2 refer to the neutron and the probe particle, respectively. The state (3) is entangled and if the two states for particle 2 are orthogonal, no interference for particle 1 can arise. Yet, if particle 2 is measured such that this measurement is not able, even in principle, to reveal any information about the slit particle 1 passes, then particle 1 will show interference. Obviously, there is a continuous transition between these two extreme situations. We thus have seen that one can either observe a wavelike feature (the interference pattern) or a particle Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999
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(3)
feature (the path a particle takes through the apparatus) depending on which experiment one chooses. Yet one could still have a naive picture in one's mind essentially assuming waves propagating through the apparatus which can only be observed in quanta. That such a picture is not possible is demonstrated by two-particle interferences, as we will discuss now. III. A DOUBLE SLIT AND TWO PARTICLES
The situation is strikingly illustrated if one employs pairs of particles which are strongly correlated ("en-
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Anton Zeilinger: Experiment and quantum physics
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FIG. 2. A source emits pairs of particles with total zero momentum. Particle 1 is either emitted into beams a or a' and particle 2 into beams b or b' with perfect correlations between a and b and a' and b', respectively. The beams of particle 1 then pass a double-slit assembly. Because of the perfect correlation between the two particles, particle 2 can serve to find out which slit particle 1 passed and therefore no interference pattern arises. tangled") such that either particle carries information about the other (Home and Zeilinger, 1985; Greenberger, Home, and Zeilinger, 1993). Consider a setup where a source emits two particles with antiparallel momenta (Fig. 2). Then, whenever particle 1 is found in beam a, particle 2 is found in beam b and whenever particle 1 is found in beam a\ particle 2 is found in beam b'. The quantum state is (4)
l=-(|a>il&>2+|a').i|fr'>2)-
Will we now observe an interference pattern for particle 1 behind its double slit? The answer has again to be negative because by simply placing detectors in the beams b and b' of particle 2 we can determine which path particle 1 took. Formally speaking, the states \a)t and \a')\ again cannot be coherently superposed because they are entangled with the two orthogonal states |&')2and|*'>2. Obviously, the interference pattern can be obtained if one applies a so-called quantum eraser which completely erases the path information carried by particle 2. That is, one has to measure particle 2 in such a way that it is not possible, even in principle, to know from the measurement which path it took, a' or b'.
A recent experiment (Dopfer, 1998) used the socalled process of parametric down conversion to create entangled pairs of photons (Fig. 3) where a UV beam entering a nonlinear optical crystal spontaneously creates pairs of photons such that the sum of their linear momenta is constant. In type-I parametric down conversion, the two photons carry equal polarization. Parametric down conversion is discussed in somewhat more detail below. Although the experimental situations are different, conceptually this is equivalent to the case discussed above. In this experiment, photon 2 passes a double slit while the other, photon 1, can be observed by a detector placed at various distances behind the Heisenberg lens which plays exactly the same role as the lens in the gamma-ray microscope discussed by Heisenberg (1928) and extended by Weizsacher (1931). If the detector is placed at the fopa! plane of the lens, then registration of a photon there provides information about its direction, i.e., momentum, before entering the lens. Thus, because of the strict momentum correlation, the momentum of the other photon incident on the double slit and registered in coincidence is also well defined. A momentum eigenstate cannot carry any position information, i.e., no information about which slit the particle passes through. Therefore, a double-slit interference pattern for photon 2 is registered conditioned on registration of photon 1 in the focal plane of the lens. It is important to note that it is actually necessary to register photon 1 at the focal plane because without registration one could always, at least in principle, reconstruct the state in front of the lens. Most strikingly, therefore, one can find out the slit photon 2 passed by placing the detector for photon 1 into the imaging plane of the lens. The imaging plane is simply obtained by taking the object distance as the sum of the distances from the lens to the crystal and from the crystal to the double slit. Then, as has also been demonstrated in the experiment, a one-to-one relationship exists between positions in the plane of the double slit and in the imaging plane and thus, the slit particle 2 passes through can readily be determined by observing photon 1 in the imaging plane. Only after registration of photon 1 in the
Heisenberg Lens
Heisenberg Detector
UV-Pump Crystal
Double Slit Detector
Coincidence Logic
FIG, 3. Two photons and one double slit. A pair of momentum-entangled photons is created by type-/ parametric down conversion. Photon 2 enters a double-slit assembly and photon 1 is registered by the Heisenberg detector arranged behind the Heisenberg lens. If the Heisenberg detector is placed in the focal plane of the lens, it projects the state of the second photon into a momentum eigenstate which cannot reveal any position information and hence no information about slit passage. Therefore, in coincidence with a registration of photon 1 in the focal plane, photon 2 exhibits an interference pattern. On the other hand, if the Heisenberg detector is placed in the imaging plane at 2 /, it can reveal the path the second photon takes through the slit assembly which therefore connot show the interference pattern (Dopfer, 1998). Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999
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One might still be tempted to assume a picture that the source emits a statistical mixture of pairwise correlated waves where measurement of one photon just selects a certain, already existing, wavelet for the other photon. It is easy to see that any such picture cannot lead to the perfect interference modulation observed. The most sensible position, according to quantum mechanics, is to assume that no such waves preexist before any measurement. IV. QUANTUM COMPLEMENTARITY
-8000
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0
2000
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Position of Heisenberg-Detector D1 [jim]
FIG. 4. Double-slit pattern registered by the Heisenberg detector of photon 1 (Fig. 3). The graph shows the counts registered by that detector as a function of its lateral position, if that detector is arranged in the focal plane of the lens. The counts are conditioned on registration of the second photon behind its double slit. Note that the photons registered in detector D1 exhibit a double-slit pattern even though they never pass through a double-slit assembly. Note also the low intensity which indicates that the interference pattern is collected photon by photon. focal plane of the lens is any possibility to obtain any path information from photon 1 irrecoverably destroyed. We note that the distribution of photons behind the double slit without registration of the other photon is just an incoherent sum of probabilities having passed through either slit and, as shown in the experiment, no interference pattern arises if one does not look at the other photon. This is again a result of the fact that, indeed, path information is still present and can easily be extracted by placing the detector of photon 1 into the imaging plane of the lens. Likewise, registration of photon 2 behind its double slit destroys any path information it may carry and thus, by symmetry, a Fraunhofer double-slit pattern is obtained for the distribution of photon 1 in the focal plane behind its lens, even though that photon never passed a double slit (Fig. 4)! This experiment can be understood intuitively if we carefully analyze what registration of a photon behind a double slit implies. It simply means that the state incident on the double slit is collapsed into a wave packet with the appropriate momentum distribution such that the wave packet peaks at both slits. By virtue of the strong momentum entanglement at the source, the other wave packet then has a related momentum distribution which actually is, according to an argument put forward by Klyshko (1988), the time reversal of the other wave packet. Thus, photon 1 appears to originate backwards from the double slit assembly and is then considered to be reflected by the wave fronts of the pump beam into the beam towards the lens which then simply realizes the standard Fraunhofer observation conditions. Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999
The observation that particle path and interference pattern mutually exclude each other is one specific manifestation of the general concept of complementary in quantum physics. Other examples are position and linear momentum as highlighted in Heisenberg's uncertainty relation, or the different components of angular momentum. It is often said that complementarity is due to an unavoidable disturbance during observation. This is suggested if, as in our example in Sec. II, we consider determining the path a particle takes through the double-slit assembly by scattering some other particle from it. That this is too limited a view is brought out by the experiment discussed in the preceding section. The absence of the interference pattern for photon 2 if no measurement is performed on photon 1, is not due to it being disturbed by observation; rather, it can be understood if we consider the complete set of possible statements which can be made about the experiment as a whole (Bohr, 1935) including the other photon. As long as no observation whatsoever is made on the complete quantum system comprised of both photons our description of the situation has to encompass all possible experimental results. The quantum state is exactly that representation of our knowledge of the complete situation which enables the maximal set of (probabilistic) predictions for any possible future observation. What comes new in quantum mechanics is that, instead of just listing the various experimental possibilities with the individual probabilities, we have to represent our knowledge of the situation by the quantum state using complex amplitudes. If we accept that the quantum state is no more than a representation of the information we have, then the spontaneous change of the state upon observation, the so-called collapse or reduction of the wave packet, is just a very natural consequence of the fact that, upon observation, our information changes and therefore we have to change our representation of the information, that is, the quantum state. From that position, the so-called measurement problem (Wigner, 1970) is not a problem but a consequence of the more fundamental role information plays in quantum physics as compared to classical physics (Zeilinger, 1999). Quantum complementarity then is simply an expression of the fact that in order to measure two complementary quantities, we would have to use apparatuses which mutually exclude each other. In the example of our experiment, interference pattern and path information for photon 2 are mutually exclusive, i.e., comple-
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mentary, because ft is not possible to position the detector for photon 1 simultaneously in the focal plane and in the image plane of the lens. Yet the complete quantum state encompasses both possible experiments. We finally note two corollaries of our analysis. First, it is clearly possible to have a concept of continuous complementarity. In our case, placing the detector of photon 1 somewhere in between the two extreme positions mentioned will reveal partial path information and thus an interference pattern of reduced visibility. And second, the choice whether or not path information or the interference pattern become manifest for photon 2 can be delayed to arbitrary times after that photon has been registered. In the experiment discussed, the choice where detector Dt is placed can be delayed until after photon 2 has been detected behind its double slit. While we note that in the experiment, the lens was already arranged at a larger distance from the crystal than the double slit, a future experiment will actually employ a rapidly switched mirror sending photon 1 either to a detector placed in the focal plane of the lens or to a detector placed in the imaging plane. This possibility of deciding long after registration of the photon whether a wave feature or a particle feature manifests itself is another warning that one should not have any realistic pictures in one's mind when considering a quantum phenomenon. Any detailed picture of what goes on in a specific individual observation of one photon has to take into account the whole experimental apparatus of the complete quantum system consisting of both photons and it can only make sense after the fact, i.e., after all information concerning complementary variables has irrecoverably been erased.
V. EINSTEIN-PODOLSKY-ROSEN AND BELL'S INEQUALITY
In 1935 Einstein, Podolsky, and Rosen (EPR) studied entangled states of the general type used in the twophoton experiment discussed above. They realized that in many such states, when measuring either linear momentum or position of one of the two particles, one can infer precisely either momentum or position of the other. As the two particles might be widely separated, it is natural to assume validity of the locality condition suggested by EPR: "Since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system." Then, whether or not momentum or position can be assigned to particle (system) 2 must be independent of what measurement is performed on particle 1 or even whether any measurement is performed on it at all. The question therefore arises whether the specific results obtained for either particle can be understood without reference to which measurement is actually performed on the other particle. Such a picture would imply a theory, underlying quantum physics, which provides a more detailed acRev. Mod. Phys., Vol. 71, No. 2, Centenary 1999
FIG. 5. Typical experimental arrangement to test Bell's inequality. A source emits, say, polarization-entangled pairs of photons. Each photon is sent through a polarizer whose orientation can be varied. Finally behind each polarizer, the transmitted photons are registered. Quantum mechanics predicts a sinusoidal variation of the coincidence count rate as a function of the relative angular orientation of the polarizers. Any such variation violates local realism as expressed in Bell's inequality. count of individual measurements. Specifically, following Bell, it might explain "why events happen" (Bell, 1990; Gottfried, 1991). In the sixties, two different developments started, which nicely complement each other. First, it was initially argued by Specker (1960) for Hibbert spaces of dimension larger than two that quantum mechanics cannot be supplemented by additional variables. Later it was shown by Kochen and Specker (1967) and by Bell (1966; for a review see Mermin, 1993), that for the specific case of a spin-1 particle, it is not possible to assign in a consistent way measurement values to the squares of any three orthogonal spin projections, despite the fact that the three measurements commute with each other. This is a purely geometric argument which only makes use of some very basic geometric considerations. The conclusion here is very important. The quantum system cannot be assigned properties independent of the context of the complete experimental arrangement. This is just in the spirit of Bohr's interpretation. This so-called contextuality of quantum physics is another central and subtle feature of quantum mechanics. Second, a most important development was due to John Bell (1964) who continued the EPR line of reasoning and demonstrated that a contradiction arises between the EPR assumptions and quantum physics. The most essential assumptions are realism and locality. This contradiction is called Bell's theorem. To be specific, and in anticipation of experiments we will discuss below, let us assume we have a pair of photons in the state: l«A)=^(l^)1|^)2-|l/)il^>2)-
(5)
This polarization-entangled state implies that whenever (Fig. 5) photon 1 is measured and found to have horizontal (H) polarization, the polarization of photon 2 will be vertical (V) and vice versa. Actually, the state of Eq. (5) has the same form in any basis. This means whichever state photon 1 will be found in, photon 2 can definitely be predicted to be found in the orthogonal state if measured.
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Following EPR one can apply their famous reality criterion, "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." This would imply that to any possible polarization measurement on any one of our photons we can assign such an element of physical reality on the basis of a corresponding measurement on the other photon of any given pair. The next step then is to assume the two photons (systems) to be widely separated so that we can invoke EPR's locality assumption as given above. Within this line of reasoning, whether or not we can assign an element of reality to a specific polarization of one of the systems must be independent of which measurement we actually perform on the other system and even independent of whether we care to perform any measurement at all on that system. To put it dramatically, one experiment could be performed here on earth and the other on a planet of another star a couple of light years away. It is this very independence of a measurement result on one side from what may be done on the other side, as assumed by EPR, which is at variance with quantum mechanics. Indeed, this assumption implies that certain combinations of expectation values have definite bounds. The mathematical expression of that bound is called Bell's inequality, of which many variants exist. For example, a version given by Clauser, Home, Shimony, and Holt (1969) is \E(a,p)-E(a'
,p)\ + \E(a,p')
+E(a' ,/3')|=s2, (6)
where
E(a,p)=jj[C + + (a,P) + C—( Richard Jozsa, ( 2 ) Asher Peres, ( 4 ) and William K. Wootters ( 6 ' ^'IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York 10598 (2) Di.partem.ent IRO, University de Montreal, C.P. 6128, Succursale "A", Montreal, Quebec, Canada HSC SJ7 (3 'Laboratoire d'Informatique de I'^cole Normale Supirieure, 45 rue d'Ulm, 752S0 Paris CEDBX 05, Prance^ (4) Department of Physics, Technion-Israel Institute of Technology, SS000 Haifa, Israel W Department of Physics, Williams College, Williamstown, Massachusetts 01267 (Received 2 December 1992) An unknown quantum state |0) can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, "Alice," and the receiver, "Bob," must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state \(j>) which Alice destroyed. PACS numbers: 03.65.Bz, 42.50.Dv, 89.70.+C The existence of long range correlations between Einstein-Podolsky-Rosen (EPR) [1] pairs of particles raises the question of their use for information transfer. Einstein himself used t h e word "telepathically" in this context [2]. It is known t h a t instantaneous information transfer is definitely impossible [3]. Here, we show t h a t E P R correlations can nevertheless assist in the "teleportation" of an intact quantum state from one place to another, by a sender who knows neither the state to be teleported nor the location of the intended receiver. Suppose one observer, whom we shall call "Alice," has been given a quantum system such as a photon or spin-5 particle, prepared in a state \d>) unknown to her, and she wishes to communicate to another observer, "Bob," sufficient information about t h e quantum system for him to make an accurate copy of it. Knowing t h e state vector |0) itself would be sufficient information, but in general there is no way t o learn it. Only if Alice knows beforehand t h a t \d>) belongs to a given orthonormal set can she make a measurement whose result will allow her to make an accurate copy of \<j>). Conversely, if t h e possibilities for \) include two or more nonorthogonal states, then no measurement will yield sufficient information to prepare
a perfectly accurate copy. A trivial way for Alice to provide Bob with all the information in \<j>) would be to send the particle itself. If she wants to avoid transferring the original particle, she can make it.interact unitarily with another system, or "ancilla," initially in a known state \ao), in such a way t h a t after the interaction the original particle is left in a standard state \o) and the ancilla is in an unknown state I a) containing complete information about \cf>). If Alice now sends Bob the ancilla (perhaps technically easier than sending the original particle), Bob can reverse her actions to prepare a replica of her original state \4>). This "spin-exchange measurement" [4] illustrates an essential feature of quantum information: it can be swapped from one system to another, but it cannot be duplicated or "cloned" [5]. In this regard it is quite unlike classical information, which can be duplicated at will. The most tangible manifestation of t h e nonclassicality of quantum information is the violation of Bell's inequalities [6] observed [7] in experiments on E P R states. Other manifestations include the possibility of quantum cryptography [8], quantum parallel computation [9], and the superiority of interactive measurements for extracting informa-
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tion from a pair of identically prepared particles [10]. The spin-exchange method of sending full information to Bob still lumps classical and nonclassical information together in a single transmission. Below, we show how Alice can divide the full information encoded in \). An entanglement between these two subsystems is brought about in the next step. To couple the first particle with the E P R pair, Alice performs a complete measurement of the von Neumann type on the joint system consisting of particle 1 and particle 2 (her E P R particle). This measurement is performed in the Bell operator basis [11] consisting of l ^ ' j ) and l*l!2))=\/|(ITl>U2)+Ul)IT2», (2)
l*(1f)>
= \/?(ITi)IT 2 )±U 1 >ll2».
Note that these four states are a complete orthonormal basis for particles 1 and 2. It is convenient to write the unknown state of the first particle as
l*i> = «*ITi>+&Ui>,
(3)
with |a| 2 + |6| 2 = 1. The complete state of the three particles before Alice's measurement is thus
The subscripts 2 and 3 label the particles in this E P R pair. Alice's original particle, whose unknown state \cj>) l*iH> = ^ g ( | T i > | T a > | l a > - | T i > | i a > I T s » she seeks to teleport to Bob, will be designated by a subscript 1 when necessary. These three particles may be +^|(lii>lTa>U8>-Ui>Ua>IT3». (4) of different kinds, e.g., one or more may be photons, the polarization degree of freedom having the same algebra In this equation, each direct product | i ) | 2) can be exas a spin. One E P R particle (particle 2) is given to Alice, while pressed in terms of the Bell operator basis vectors \$>\i ) and 1^12 ), and we obtain I*i23> = * fl*^) ( - a | f ,> - b\ | 3 ) ) -r-1*) ( - a | Ts> • &Ua» + \*tf)
Wis) +6|Ts» + ]${?) H i s ) -6|T»»]. (5)
It follows that, regardless of the unknown state \<j>i), the four measurement outcomes are equally likely, each occurring with probability 1/4. Furthermore, after Alice's measurement, Bob's particle 3 will have been projected into one of the four pure states superposed in Eq. (5), according to the measurement outcome. These are, respectively,
- w - ( ! ) . (1!)i*>. (6)
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Each of these possible resultant states for Bob's EPR particle is related in a simple way to the original state \) which Alice sought to teleport. In the case of the first (singlet) outcome, Bob's state is the same except for an irrelevant phase factor, so Bob need do nothing further to produce a replica of Alice's spin. In the three other cases, Bob must apply one of the unitary operators in Eq. (6), corresponding, respectively, to 180° rotations around the z, x, and y axes, in order to convert his E P R particle into a replica of Alice's original state \<j>). (If \<j>) represents a photon polarization state, a suitable combination of half-
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wave plates will perform these unitary operations.) Thus an accurate teleportation can be achieved in all cases by having Alice tell Bob the classical outcome of her measurement, after which Bob applies the required rotation to transform the state of his particle into a replica of \4>). Alice, on the other hand, is left with particles 1 and 2 in one of the states |). Unlike the quantum correlation of Bob's E P R particle 3 to Alice's particle 2, the result of Alice's measurement is purely classical information, which can be transmitted, copied, and stored at will in any suitable physical medium. In particular, this information need not be destroyed or canceled to bring the teleportation process to a successful conclusion: T h e teleportation of \) from Alice to Bob has t h e side effect of producing two bits of random classical information, uncorrelated t o \cj>), which are left behind at the end of t h e process. Since teleportation is a linear operation applied t o t h e quantum state \(j>), it will work not only with pure states, but also with mixed or entangled states. For example, let Alice's original particle 1 be itself part of an E P R singlet with another particle, labeled 0, which may be far away from both Alice and Bob. Then, after teleportation, particles 0 and 3 would be left in a singlet state, even though they had originally belonged to separate E P R pairs. All of what we have said above can be generalized to systems having N > 2 orthogonal states. In place of an E P R spin pair in t h e singlet state, Alice would use a pair of AT-state particles in a completely entangled state. For definiteness let us write this entangled state as E j \J) ® \J)/VN, where j = 0 , 1 , . . . , TV - 1 labels the N elements of an orthonormal basis for each of the AT-state systems. As before, Alice performs a joint measurement on particles 1 and 2. One such measurement t h a t has the desired effect is t h e one whose eigenstates are I V w ) , defined by
IVW) = J2 e^iJn/N
\J) ® 10' + m) mod N) / VW.
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\4>) will be reconstructed (in the s p i n - j case) as a random mixture of the four states of Eq. (6). For any \), this is a maximally mixed state, giving no information about the input state \<j>). It could not be otherwise, because any correlation between the input and the guessed output could be used to send a superluminal signal. One may still inquire whether accurate teleportation of a two-state particle requires a full two bits of classical information. Could it be done, for example, using only two or three distinct classical messages instead of four, or four messages of unequal probability? Later we show that a full two bits of classical channel capacity are necessary. Accurate teleportation using a classical channel of any lesser capacity would allow Bob to send superluminal messages through the teleported particle, by guessing the classical message before it arrived (cf. Fig. 2). Conversely one may inquire whether other states besides an E P R singlet can be used as the nonclassical channel of t h e teleportation process. Clearly any direct product state of particles 2 and 3 is useless, because for such states manipulation of particle 2 has no effect on what can be predicted about particle 3. Consider now a nonfactorable state IT23) • It can readily be seen t h a t after Alice's measurement, Bob's particle 3 will be related to \4>i) by four fixed unitary operations if and only if IT23) has the form "2) |?3> + \v2) \q3)),
(9)
where {|u>, \v)} and {|p),|g)} are any two pairs of orthonormal states. These are maximally entangled states [11], having maximally random marginal statistics for measurements on either particle separately. States which are less entangled reduce the fidelity of teleportation, and/or the range of states \) t h a t can be accurately teleported. T h e states in Eq. (9) are also precisely those obtainable from the E P R singlet by a local one-particle unitary operation [12]. Their use for t h e nonclassical channel is entirely equivalent to t h a t of the singlet (1). Maximal entanglement is necessary and sufficient for faithful tele-
3
(7)
Two bits
I©
Two bits
Once Bob learns from Alice t h a t she has obtained the result nm, he performs on his previously entangled particle (particle 3) the unitary transformation Unm = J2 e2*ikn'N
\k)((k + m) mod iV|.
(8)
k
This transformation brings Bob's particle to the original state of Alice's particle 1, and the teleportation is complete. T h e classical message plays an essential role in teleportation. To see why, suppose t h a t Bob is impatient, and tries to complete t h e teleportation by guessing Alice's classical message before it arrives. Then Alice's expected
FIG. 1. Spacetime diagrams for (a) quantum teleportation, and (b) 4-way coding [12]. As usual, time increases from bottom to top. The solid lines represent a classical pair of bits, the dashed lines an EPR pair of particles (which may be of different types), and the wavy line a quantum particle in an unknown state \<j>). Alice (A) performs a quantum measurement, and Bob (B) a unitary operation. 1897
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FIG. 2. Spacetime diagram of a more complex 4-way coding scheme in which the modulated EPR particle (wavy line) is teleported rather than being transmitted directly. This diagram can be used to prove that a classical channel of two bits of capacity is necessary for teleportation. To do so, assume on the contrary that the teleportation from A' to B' uses an internal classical channel of capacity C < 2 bits, but is still able to transmit the wavy particle's state accurately from A' to B', and therefore still transmit the external two-bit message accurately from B to A. The assumed lower capacity C < 2 of the internal channel means that if B' were to guess the internal classical message superluminally instead of waiting for it to arrive, his probability 2~c of guessing correctly would exceed 1/4, resulting in a probability greater than 1/4 for successful superluminal transmission of the external twobit message from B to A. This in turn entails the existence of two distinct external two-bit messages, r and a, such that P(,r\s), the probability of superluminally receiving r if a was sent, is less than 1/4, while P(r\r), the probability of superluminally receiving r if r was sent, is greater than 1/4. By redundant coding, even this statistical difference between r and a could be used to send reliable superluminal messages; therefore reliable teleportation of a two-state particle cannot be achieved with a classical channel of less than two bits of capacity. By the same argument, reliable teleportation of an iV-state particle requires a classical channel of 21og2(iV) bits capacity. portation. Although it is currently unfeasible to store separated E P R particles for more than a brief time, if it becomes feasible to do so, quantum teleportation could be quite useful. Alice and Bob would only need a stockpile of E P R pairs (whose reliability can be tested by violations of Bell's inequality [7]) and a channel capable of carrying robust classical messages. Alice could then teleport quantum states to Bob over arbitrarily great distances, without worrying about the effects of attenuation and noise on, say, a single photon sent through a long optical fiber. As an application of teleportation, consider the problem investigated by Peres and Wootters [10], in which Bob already has another copy of \). If he acquires Alice's copy, he can measure both together, thereby determining the state \<j>) more accurately than can be done by making a separate measurement on each one. Finally, teleportation has the advantage of still being possible in situations where Alice and Bob, after sharing their E P R pairs, have wandered about independently and no longer know each others' locations. Alice cannot reliably send 1898
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Bob the original quantum particle, or a spin-exchanged version of it, if she does not know where he is; but she can still teleport the quantum state to him, by broadcasting the classical information to all places where he might be. Teleportation resembles another recent scheme for using E P R correlations to help transmit useful information. In "4-way coding" [12] modulation of one member of an E P R pair serves to reliably encode a 2-bit message in the joint state of the complete pair. Teleportation and 4-way coding can be seen as variations on the same underlying process, illustrated by the spacetime diagrams in Fig. 1. Note t h a t closed loops are involved for both processes. Trying to draw similar "Peynman diagrams" with tree structure, rather than loops, would lead to physically impossible processes. On the other hand, more complicated closed-loop diagrams are possible, such as Fig. 2, obtained by substituting Fig. 1(a) into the wavy line of Fig. 1(b). This represents a 4-way coding scheme in which the modulated E P R particle is teleported instead of being transmitted directly. Two incoming classical bits on the lower left are reproduced reliably on the upper right, with the assistance of two shared E P R pairs and two other classical bits, uncorrelated with the external bits, in an internal channel from A' to B'. This diagram is of interest because it can be used to show t h a t a full two bits of classical channel capacity are necessary for accurate teleportation of a two-state particle (cf. caption). Work by G.B. is supported by NSERC's E. W. R. Steacie Memorial Fellowship and Quebec's FCAR. A.P. was supported by the Gerard Swope Fund and the Fund for Encouragement of Research. Laboratoire dTnformatique de l'Ecole Normale Superieure is associee au CNRS URA 1327.
'*' Permanent address. [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935); D. Bohm, Quantum Theory (Prentice Hall, Englewood Cliffs, NJ, 1951). [2] A. Einstein, in Albert Einstein, Philosopher-Scientist, edited by P. A. Schilpp (Library of Living Philosophers, Evanston, 1949) p. 85. (3] A. Shimony, in Proceedings of the International Symposium on Foundations of Quantum Theory (Physical Society of Japan, Tokyo, 1984). [4] J. L. Park, Pound. Phys. 1, 23 (1970). [5] W. K. Wootters and W. H. Zurek, Nature (London) 299, 802 (1982). [6] J. S. Bell, Physics (Long Island City, N.Y.) 1,195 (1964); J. F. Clauser, M. A. Home, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). [7] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982); Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 6 1 , 2921 (1988). [8] S. Wiesner, Sigact News 15, 78 (1983); C. H. Bennett and G. Brassard, in Proceedings of IEEE International
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Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 1984), pp. 175179; A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991); C. H. Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Lett. 68, 557-559 (1992); C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992); A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, Phys. Rev. Lett. 69, 1293 (1992); C. H. Bennett, G. Brassard, C. Crepeau, and M.H. Skubiszewska, Advances in Cryptology—Crypto '91 Proceedings, August 1991 (Springer, New York, 1992), pp. 351-366; G. Brassard and C. Crepeau, Advances in Cryptology—Crypto '90 Proceedings, August 1990 (Springer, New York, 1991), pp. 49-61. [9] D. Deutsch Proc. R. Soc. London A 400, 97 (1985);
LETTERS
29 M A R C H 1993
D. Deutsch and R. Jozsa, Proc. R. Soc. London A 439, 553-558.(1992); A. Berthiaume and G. Brassard, in Proceedings of the Seventh Annual IEEE Conference on Structure in Complexity Theory, Boston, June 1992, (IEEE, New York, 1989), pp. 132-137; "Oracle Quantum Computing," Proceedings of the Workshop on Physics and Computation, PhysComp 92, (IEEE, Dallas, to be published). [10] A. Peres and W. K. Wootters, Phys. Rev. Lett. 66, 1119 (1991). [11] S. L. Braunstein, A. Mann, and M. Revzen, Phys. Rev. Lett. 68, 3259 (1992). [12] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
1899
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Experimental quantum teleportation Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter & Anton Zeilinger Institutfiir Experimentalphysik, Universitat Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
Quantum teleportation-the transmission and reconstruction over arbitrary distances of the state of a quantum system-is demonstrated experimentally. During teleportation, an initial photon which carries the polarization that is to be transferred and one of a pair of entangled photons are subjected to a measurement such that the second photon of the entangled pair acquires the polarization of the initial photon. This latter photon can be arbitrarily far away from the initial one. Quantum teleportation will be a critical ingredient for quantum computation networks. The dream of teleportation is to be able to travel by simply reappearing at some distant location. An object to be teleported can be fully characterized by its properties, which in classical physics can be determined by measurement. To make a copy of that object at a distant location one does not need the original parts and pieces—all that is needed is to send the scanned information so that it can be used for reconstructing the object. But how precisely can this be a true copy of the original? What if these parts and pieces are electrons, atoms and molecules? What happens to their individual quantum properties, which according to the Heisenberg's uncertainty principle cannot be measured with arbitrary precision? Bennett et aV have suggested that it is possible to transfer the quantum state of a particle onto another particle—the process of quantum teleportation—provided one does not get any information about the state in the course of this transformation. This requirement can be fulfilled by using entanglement, the essential feature of quantum mechanics2. It describes correlations between quantum systems much stronger than any classical correlation could be. The possibility of transferring quantum information is one of the cornerstones of the emerging field of quantum communication and quantum computation 3 . Although there is fast progress in the theoretical description of quantum information processing, the difficulties in handling quantum systems have not allowed an equal advance in the experimental realization of the new proposals. Besides the promising developments of quantum cryptography4 (the first provably secure way to send secret messages), we have only recently succeeded in demonstrating the possibility of quantum dense coding3, a way to quantum mechanically enhance data compression. The main reason for this slow experimental progress is that, although there exist methods to produce pairs of entangled photons 6 , entanglement has been demonstrated for atoms only very recently7 and it has not been possible thus far to produce entangled states of more than two quanta. Here we report the first experimental verification of quantum teleportation. By producing pairs of entangled photons by the process of parametric down-conversion and using two-photon interferometry for analysing entanglement, we could transfer a quantum property (in our case the polarization state) from one photon to another. The methods developed for this experiment will be of great importance both for exploring the field of quantum communication and for future experiments on the foundations of quantum mechanics.
and she wants Bob, at a distant location, to have a particle in that state. There is certainly the possibility of sending Bob the particle directly. But suppose that the communication channel between Alice and Bob is not good enough to preserve the necessary quantum coherence or suppose that this would take too much time, which could easily be the case if I \f) is the state of a more complicated or massive object. Then, what strategy can Alice and Bob pursue? As mentioned above, no measurement that Alice can perform on I \f) will be sufficient for Bob to reconstruct the state because the state of a quantum system cannot be fully determined by measurements. Quantum systems are so evasive because they can be in a superposition of several states at the same time. A measurement on the quantum system will force it into only one of these states—this is often referred to as the projection postulate. We can illustrate this important quantum feature by taking a single photon, which can be horizontally or vertically polarized, indicated by the states I +011>
(1)
where a and/3 are two complex numbers satisfying |o;|2 + |/3|2 = 1. To place this example in a more general setting we can replace the states I *->} and 11) in equation (1) by 10) and 11), which refer to the states of any two-state quantum system. Superpositions of 10) and I l) are called qubits to signify the new possibilities introduced by quantum physics into information science8. If a photon in state I \p) passes through a polarizing beamsplitter—a device that reflects (transmits) horizontally (vertically) polarized photons—it will be found in the reflected (transmitted) beam with probability I a 12 (I /312). Then the general state 1 \j/) has been projected either onto I«-») or onto 11) by the action of the measurement. We conclude that the rules of quantum mechanics, in particular the projection postulate, make it impossible for Alice to perform a measurement on 1if) by which she would obtain all the information necessary to reconstruct the state.
The concept of quantum teleportation Although the projection postulate in quantum mechanics seems to bring Alice's attempts to provide Bob with the state I if) to a halt, it was realised by Bennett et al.' that precisely this projection postulate enables teleportation of I if) from Alice to Bob. During teleportation Alice will destroy the quantum state at hand while Bob receives the quantum state, with neither Alice nor Bob obtaining information about the state I if). A key role in the teleportation scheme is played The problem To make the problem of transferring quantum information clearer, by an entangled ancillary pair of particles which will be initially suppose that Alice has some particle in a certain quantum state 1 ip) shared by Alice and Bob. NATURE I VOL 390111 DECEMBER 1997
575
articles Suppose particle 1 which Alice wants to teleport is in the initial far away? Einstein, among many other distinguished physicists, state |i/»), = a] «-»), + @\ J), (Fig. la), and the entangled pair of could simply not accept this "spooky action at a distance". But this particles 2 and 3 shared by Alice and Bob is in the state: property of entangled states has now been demonstrated by numerous experiments (for reviews, see refs 9, 10). The teleportation scheme works as follows. Alice has the particle 1 hr> B =4=(iHitf>3-it>2Mb) (2) in the initial state I \j/)i and particle 2. Particle 2 is entangled with V2 That entangled pair is a single quantum system in an equal particle 3 in the hands of Bob. The essential point is to perform a superposition of the states l*-*^!!^ and It^l* - *^- The entangled specific measurement on particles 1 and 2 which projects them onto state contains no information on the individual particles; it only the entangled state: indicates that the two particles will be in opposite states. The important property of an entangled pair is that as soon as a V2 measurement on one of the particles projects it, say, onto I ~)23> which means that particle 2 is also orthogonal to particle 3. This is only possible if EPR-source particle 3 is in the same state as particle 1 was initially. The final state of particle 3 is therefore: ll»3 = a | " > 3 + |3|t>3
Figure 1 Scheme showing principles involved in quantum teleportation (a) and the experimental set-up (b). a, Alice has a quantum system, particle 1, in an initial state which she wants to teleport to Bob. Alice and Bob also share an ancillary entangled pair of particles 2 and 3 emitted by an Einstein-Podolsky-Rosen(EPR) source. Alice then performs a joint Bell-state measurement (BSM) on the initial particle and one of the ancillaries, projecting them also onto an entangled state. After she has sent the result of her measurement as classical information to Bob, he can perform a unitary transformation (U) on the other ancillary particle resulting in it being in the state of the original particle, b, A pulse of ultraviolet radiation passing through a nonlinear crystal creates the ancillary pair of photons 2 and 3. After retroftection during its second passage through the crystal the ultraviolet pulse creates another pair of photons, one of which will be prepared in the initial state of photon 1 to be teleported, the other one serving as a trigger indicating that a photon to be teleported is under way. Alice then looks for coincidences after a beam splitter BS where the initial photon and one of the ancillaries are superposed. Bob, after receiving the classical information that Alice obtained a coincidence count in detectors f1 and f2 identifying the |^")i 2 Bell state, knows that his photon 3 is in the initial state of photon 1 which he then can check using polarization analysis with the polarizing beam splitter PBS and the detectors d1 and d2. The detector p provides the information that photon 1 is under way.
576
(4)
We note that during the Bell-state measurement particle 1 loses its identity because it becomes entangled with particle 2. Therefore the state I i£)i is destroyed on Alice's side during teleportation. This result (equation (4)) deserves some further comments. The transfer of quantum information from particle 1 to particle 3 can happen over arbitrary distances, hence the name teleportation. Experimentally, quantum entanglement has been shown11 to survive over distances of the order of 10 km. We note that in the teleportation scheme it is not necessary for Alice to know where Bob is. Furthermore, the initial state of particle 1 can be completely unknown not only to Alice but to anyone. It could even be quantum mechanically completely undefined at the time the Bell-state measurement takes place. This is the case when, as already remarked by Bennett era/.1, particle 1 itself is a member of an entangled pair and therefore has no well-defined properties on its own. This ultimately leads to entanglement swapping1213. It is also important to notice that the Bell-state measurement does not reveal any information on the properties of any of the particles. This is the very reason why quantum teleportation using coherent two-particle superpositions works, while any measurement on oneparticle superpositions would fail. The fact that no information whatsoever is gained on either particle is also the reason why quantum teleportation escapes the verdict of the no-cloning theorem 14 . After successful teleportation particle 1 is not available in its original state any more, and therefore particle 3 is not a clone but is really the result of teleportation. A complete Bell-state measurement can not only give the result that the two particles 1 and 2 are in the antisymmetric state, but with equal probabilities of 25% we could find them in any one of the three other entangled states. When this happens, particle 3 is left in one of three different states. It can then be brought by Bob into the original state of particle 1 by an accordingly chosen transformation, independent of the state of particle 1, after receiving via a classical communication channel the information on which of the Bell-state NATURE I VOL 390111 DECEMBER 1997
articles results was obtained by Alice. Yet we note, with emphasis, that even if we chose to identify only one of the four Bell states as discussed above, teleportation is successfully achieved, albeit only in a quarter of the cases.
Experimental realization Teleportation necessitates both production and measurement of entangled states; these are the two most challenging tasks for any experimental realization. Thus far there are only a few experimental techniques by which one can prepare entangled states, and there exist no experimentally realized procedures to identify all four Bell states for any kind of quantum system. However, entangled pairs of photons can readily be generated and they can be projected onto at least two of the four Bell states. We produced the entangled photons 2 and 3 by parametric downconversion. In this technique, inside a nonlinear crystal, an incoming pump photon can decay spontaneously into two photons which, in the case of type II parametric down-conversion, are in the state given by equation (2) (Fig. 2) 6 . To achieve projection of photons 1 and 2 into a Bell state we have to make them indistinguishable. To achieve this indistinguishability we superpose the two photons at a beam splitter (Fig. lb). Then if they are incident one from each side, how can it happen that they emerge still one on each side? Clearly this can happen if they are either both relected or both transmitted. In quantum physics we have to superimpose the amplitudes for these two possibilities. Unitarity implies that the amplitude for both photons being relected obtains an additional minus sign. Therefore, it seems that the two processes cancel each other. This is, however, only true for a symmetric input state. For an antisymmetric state, the two possibilities obtain another relative minus sign, and therefore they constructively interfere15*16. It is thus sufficient for projecting photons 1 and 2 onto the antisymmetric state I t£_)12 to place detectors in each of the outputs of the beam splitter and to register simultaneous detections (coincidence)17"19. To make sure that photons 1 and 2 cannot be distinguished by their arrival times, they were generated using a pulsed pump beam and sent through narrow-bandwidth filters producing a coherence time much longer than the pump pulse length20. In the experiment,
the pump pulses had a duration of 200 fs at a repetition rate of 76 MHz. Observing the down-converted photons at a wavelength of 788 nm and a bandwidth of 4 nm results in a coherence time of 520 fs. It should be mentioned that, because photon 1 is also produced as part of an entangled pair, ife partner can serve to indicate that it was emitted. How can one experimentally prove that an unknown quantum state can be teleported? First, one has to show that teleportation works for a (complete) basis, a set of known states into which any other state can be decomposed. A basis for polarization states has just two components, and in principle we could choose as the basis horizontal and vertical polarization as emitted by the source. Yet this would not demonstrate that teleportation works for any general superposition, because these two directions are preferred directions in our experiment. Therefore, in the first demonstration we choose as the basis for teleportation the two states linearly polarized at -45° and +45° which are already superpositions of the horizontal and vertical polarizations. Second, one has to show that teleportation works for superpositions of these base states. Therefore we also demonstrate teleportation for circular polarization.
Results In the first experiment photon 1 is polarized at 45°. Teleportation should work as soon as photon 1 and 2 are detected in the I ^~} 1 2 state, which occurs in 25% of all possible cases. The I ^~) 12 state is identified by recording a coincidence between two detectors, fl and £2, placed behind the beam splitter (Fig. lb). If we detect a flf2 coincidence (between detectors fl and f2), then photon 3 should also be polarized at 45°. The polarization of photon 3 is analysed by passing it through a polarizing beam splitter selecting +45° and -45° polarization. To demonstrate teleportation, only detector d2 at the +45° output of the polarizing beam splitter should click (that is, register a detection) once detectors fl and £2 click. Detector d l at the -45° output of the polarizing beam splitter should not detect a photon. Therefore, recording a three-fold coincidence d2flf2 (+45° analysis) together with the absence of a three-fold coincidence dlfl£2 (-45° analysis) is a proof that the polarization of photon 1 has been teleported to photon 3. To meet the condition of temporal overlap, we change in small
Theory: +45° teleportation
HjtfHzontal
.P^
•...
Vertical ^H^^^*^ -100 -50
0
50
100
Delay (nm) Figure 2 Photons emerging from type II down-conversion (see text). Photograph taken perpendicular to the propagation direction. Photons are produced in pairs. A photon on the top circle is horizontally polarized while its exactly opposite partner in the bottom circle is vertically polarized. At the intersection points their polarizations are undefined; all that is known is that they have to be different, which results in entanglement. NATURE I VOL 390111 DECEMBER 1997
Figure 3 Theoretical prediction for the three-fold coincidence probability between the two Bell-state detectors (f1, f2) and one of the detectors analysing the teleported state. The signature of teleportation of a photon polarization state at +45° is a dip to zero at zero delay in the three-fold coincidence rate with the detector analysing -45° (d1f1f2) (a) and a constant value for the detector analysis +45° (d2f1f2) (b). The shaded area indicates the region of teleportation. §77
articles steps the arrival time of photon 2 by changing the delay between the first and second down-conversion by translating the retroflection mirror (Fig. lb). In this way we scan into the region of temporal overlap at the beam splitter so that teleportation should occur. Outside the region of teleportation, photon 1 and 2 each will go either to f 1 or to f2 independent of one another. The probability of having a coincidence between fl and f2 is therefore 50%, which is twice as high as inside the region of teleportation. Photon 3 should not have a well-defined polarization because it is part of an entangled pair. Therefore, dl and d2 have both a 50% chance of receiving photon 3. This simple argument yields a 25% probability both for the -45° analysis (dlflf2 coincidences) and for the +45° analysis (d2flf2 coincidences) outside the region of teleportation. Figure 3 summarizes the predictions as a function of the delay. Successful teleportation of the +45° polarization state is then characterized by a decrease to zero in the -45° analysis (Fig. 3a), and by a constant value for the +45° analysis (Fig. 3b). The theoretical prediction of Fig. 3 may easily be understood by realizing that at zero delay there is a decrease to half in the coincidence rate for the two detectors of the Bell-state analyser, fl and f2, compared with outside the region of teleportation. Therefore, if the polarization of photon 3 were completely uncorrelated to the others the three-fold coincidence should also show this dip to half. That the right state is teleported is indicated by the fact that the dip goes to zero in Fig. 3a and that it is filled to a flat curve in Fig. 3b. We note that equally as likely as the production of photons 1, 2 and 3 is the emission of two pairs of down-converted photons by a single source. Although there is no photon coming from the first source (photon 1 is absent), there will still be a significant contribution to the three-fold coincidence rates. These coincidences have nothing to do with teleportation and can be identified by blocking the path of photon 1. The probability for this process to yield spurious two- and threefold coincidences can be estimated by taking into account the experimental parameters. The experimentally determined value
0
teleportation
Table 1 Visibility of teleportation in three fold coincidences Polarization
Visibility
+45° -45° 0° 90" Circular
0.63 0.64 0.66 0.61 0.57
± ± ± ± ±
0.02 0.02 0.02 0.02 0.02
for the percentage of spurious three-fold coincidences is 68% ± 1%. In the experimental graphs of Fig. 4 we have subtracted the experimentally determined spurious coincidences. The experimental results for teleportation of photons polarized under +45° are shown in the left-hand column of Fig. 4; Fig. 4a and b should be compared with the theoretical predictions shown in Fig. 3. The strong decrease in the -45° analysis, and the constant signal for the +45° analysis, indicate that photon 3 is polarized along the direction of photon 1, confirming teleportation. The results for photon 1 polarized at -45° demonstrate that teleportation works for a complete basis for polarization states (right-hand column of Fig. 4). To rule out any classical explanation for the experimental results, we have produced further confirmation that our procedure works by additional experiments. In these experiments we teleported photons linearly polarized at 0° and at 90°, and also teleported circularly polarized photons. The experimental results are summarized in Table 1, where we list the visibility of the dip in three-fold coincidences, which occurs for analysis orthogonal to the input polarization. As mentioned above, the values for the visibilities are obtained after subtracting the offset caused by spurious three-fold coincidences. These can experimentally be excluded by conditioning the three-fold coincidences on the detection of photon 4, which effectively projects photon 1 into a single-particle state. We have performed this fourfold coincidence measurement for the case of teleportation of the +45° and +90° polarization states, that is, for two non-orthogonal
-45° teleportation
45° teleportation
90° teleportation
:\ f^ -.
V *"
Ir/WY^ 40-
+45° 20-
-so o
so loo 150
Delay (urn)
-150 -100 -50
0
50 100 150
Delay (jim)
Figure 4 Experimental results. Measured three-fold coincidence rates d1f1f2 (-45°) and d2f1f2 (+45°) in the case that the photon state to be teleported is polarized at +45° (a and b) or at -45° (c and d). The coincidence rates are plotted as function of the delay between the arrival of photon 1 and 2 at Alice's beam splitter (see Fig. 1b). The three-fold coincidence rates are plotted after subtracting the spurious three-fold contribution (see text). These data, compared with Fig. 3, together with similar ones for other polarizations (Table 1) confirm teleportation for an arbitrary state.
578
O,
b
, , -150 -100 -50
, 0.... 50 100 150
Delay (urn)
•150 -100 -50
0
50 100 150
Delay (|xm)
Figure 5 Four-fold coincidence rates (without background subtraction). Conditioning the three-fold coincidences as shown in Fig, 4 on the registration of photon 4 (see Fig. 1b) eliminates the spurious three-fold background, a and b show the four-fold coincidence measurements for the case of teleportation of the +45° polarization state; c and d show the results forthe +90° polarization state. The visibilities, and thus the polarizations of the teleported photons, obtained without any background subtraction are 70% ± 3%. These results for teleportation of two non-orthogonal states prove that we have demonstrated teleportation of the quantum state of a single photon. NATURE I VOL 390111 DECEMBER 1997
articles states. The experimental results are shown in Fig. 5. Visibilities of 70% ± 3% are obtained for the dips in the orthogonal polarization states. Here, these visibilities are directly the degree of polarization of the teleported photon in the right state. This proves that we have demonstrated teleportation of the quantum state of a single photon. The next steps In our experiment, we used pairs of polarization entangled photons as produced by pulsed down-conversion and two-photon interferometric methods to transfer the polarization state of one photon onto another one. But teleportation is by no means restricted to this system. In addition to pairs of entangled photons or entangled atoms7,21, one could imagine entangling photons with atoms, or phonons with ions, and so on. Then teleportation would allow us to transfer the state of, for example, fast-decohering, short-lived particles, onto some more stable systems. This opens the possibility of quantum memories, where the information of incoming photons is stored on trapped ions, carefully shielded from the environment. Furthermore, by using entanglement purification 22 —a scheme of improving the quality of entanglement if it was degraded by decoherence during storage or transmission of the particles over noisy channels—it becomes possible to teleport the quantum state of a particle to some place, even if the available quantum channels are of very poor quality and thus sending the particle itself would very probably destroy the fragile quantum state. The feasibility of preserving quantum states in a hostile environment will have great advantages in the realm of quantum computation. The teleportation scheme could also be used to provide links between quantum computers. Quantum teleportation is not only an important ingredient in quantum information tasks; it also allows new types of experiments and investigations of the foundations of quantum mechanics. As any arbitrary state can be teleported, so can the fully undetermined state of a particle which is member of an entangled pair. Doing so, one transfers the entanglement between particles. This allows us not only to chain the transmission of quantum states over distances, where decoherence would have already destroyed the state completely, but it also enables us to perform a test of Bell's theorem on particles which do not share any common past, a new step in the investigation of the features of quantum mechanics. Last but not least, the discussion about the local realistic character of nature
NATURElVOL 390111 DECEMBER 1997
could be settled firmly if one used features of the experiment presented here to generate entanglement between more than two spatially separated particles23,24. • Received 16 October; accepted 18 November 1997. 1. Bennett, C. H. et at Teleporting an unknown quantum state via dual classic and Einstein -PodolskyRosen channels. Phys. Rev. Lett. 70, 1895-1899 (1993). 2. Schrodinger, E. Die gegenwartige Situation in der Quantenmechanik. Naturwissenschaften 23, 807812; 823-828; 844-849 (1935). 3. Bennett, C. H. Quantum information and computation. Phys. Today 48(10), 24-30, October (1995). 4. Bennett, C. H„ Brassard, G. & Ekert, A. K. Quantum Cryptography. Set. Am. 267(4), 50-57, October (1992). 5. Mattle, K., Weinfurter, H., Kwiat, P. G. & Zeilinger, A. Dense coding in experimental quantum communication. Phys. Rev. Lett. 76, 4656-4659 (1996). 6. Kwiat, P. G. etal. New high intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337-4341 (1995). 7. Hagley, E. et al. Generation of Einstein-Pod olsky-Rosen pairs of atoms. Phys. Rev. Lett. 79,1 -5 (1997). 8. Schumacher, B. Quantum coding. Phys. Rev. A 51, 2738-2747 (1995). 9. Ciauser, ]. F. & Shimony, A. Bell's theorem: experimental tests and implications. Rep. Prog. Phys. 41, 1881-1927 (1978). 10. Greenberger, D. M., Home, M. A. & Zeilinger, A. Multipartite interferometry and the superposition principle. Phys. Today August, 22-29 (1993). 11. Tittel, W. et al. Experimental demonstration of quantum-correlations over more than 10 kilometers. Phys. Rev. Lett, (submitted). 12. Zukowski, M., Zeilinger, A., Home, M. A. & Ekert, A. "Event-ready-detectors" Bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287-4290 (1993). 13. Bose, S., Vedral, V. & Knight, P. L. A multiparticle generalization of entanglement swapping, preprint. 14. Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802-803 (1982). 15. Loudon, R. Coherence and Quantum Optics VUeds Everly, J. H. &Mandel, L.) 703-708 (Plenum, New York, 1990). 16. Zeilinger, A., Bernstein, H. J. & Home, M. A. Information transfer with two-state two-particle quantum systems./. Mod. Optics 41, 2375-2384 (1994). 17. Weinfurter, H. Experimental Bell-state analysis. Europhys. Lett. 25, 559-564 (1994). 18. Braunstein,S.L.& Mann, A. Measurement of the Bell operator and quantum teleportation. Phys. Rev. A51.R1727-R1730 (1995). 19. Michler, M., Mattle, K., Weinfurter, H. & Zeilinger, A. Interferometric Bell-state analysis. Phys. Rev. A 53, R1209-R1212 (1996). 20. Zukowski, M., Zeilinger, A. & Weinfurter, H. Entangling photons radiated by independent pulsed sources. Ann. NY Acad. Sci. 755, 91-102 (1995). 21. Fry, E. S., Walther, T. & Li, S. Proposal for a loophole-free test of the Bell inequalities. Phys. Rev. A 52, 4381-4395 (1995). 22. Bennett, C. H. et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722-725 (1996). 23. Greenberger, D. M., Home, M. A., Shimony, A. & Zeilinger, A. Bell's theorem without inequalities. Am. J. Phys. 58, 1131-1143(1990). 24. Zeilinger, A., Home, M. A., Weinfurter, H. & Zukowski, M. Three particle entanglements from two entangled pairs. Phys. Rev. Lett. 78, 3031-3034 (1997). Acknowledgements. We thank C. Bennett, I. Cirac, J. Rarity, W. Wootters and P. Zoller for discussions, and M. Zukowski for suggestions about various aspects of the experiments. This work was supported by the Austrian Science Foundation FWF, the Austrian Academy of Sciences, the TMR program of the European Union and the US NSE Correspondence and requests for materials should be addressed to D.B. (e-mail: Dik.Bouwmeester@uibk.
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V O L U M E 80, N U M B E R 4
REVIEW
LETTERS
26 JANUARY
1998
Teleportation of Continuous Quantum Variables Samuel L. Braunstein SEECS, University of Wales, Bangor LL57 1UT, United Kingdom H. J. Kimble Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125 (Received 8 September 1997) Quantum teleportation is analyzed for states of dynamical variables with continuous spectra, in contrast to previous work with discrete (spin) variables. The entanglement fidelity of the scheme is computed, including the roles of finite quantum correlation and nonideal detection efficiency. A protocol is presented for teleporting the wave function of a single mode of the electromagnetic field with high fidelity using squeezed-state entanglement and current experimental capability. [S0031-9007(97)05114-4] PACS numbers: 03.67.-a, 03.65.Bz, 42.50.Dv Quantum mechanics offers certain unique capabilities for the processing of information, whether for computation or communication [1], A particularly startling discovery by Bennett et al. is the possibility for teleportation of a quantum state, whereby an unknown state of a spin- \ particle is transported by "Alice" from a sending station to "Bob" at a receiving terminal by conveying 2 bits of classical information [2]. The enabling capability for this remarkable process is what Bell termed the irreducible nonlocal content of quantum mechanics, namely that Alice and Bob share an entangled quantum state and exploit its nonlocal characteristics for the teleportation process. For spin-1 particles, this entangled state is a pair of spins in a Bell state as in Bohm's version of the Einstein, Podolsky, and Rosen (EPR) paradox [3] and for which Bell formulated his famous inequalities [4]. Beyond the context of dichotomic variables, Vaidman has analyzed teleportation of the wave function of a onedimensional particle in a beautiful variation of the original EPR paradox [5]. In this case, the nonlocal resource shared by Alice and Bob is the EPR state with perfect correlations in both position and momentum. The goal of this Letter is to extend Vaidman's analysis to incorporate finite (nonsingular) degrees of correlation among the relevant particles and to include inefficiencies in the measurement process. The "quality" of the resulting protocol for teleportation is quantified with the first explicit computation of the fidelity of entanglement for a process acting on an infinite dimensional Hilbert space. We further describe a realistic implementation for the quantum teleportation of states of continuous variables, where now the entangled i
WEtR(on;a2)
= ~^exp{-e —* CS(x\
+
2r
[(xi
X2)S(pi
*2) 2 + (pi
state shared by Alice and Bob is a highly squeezed twomode state of the electromagnetic field, with the quadrature amplitudes of the field playing the roles of position and momentum. Indeed, an experimental demonstration of the original EPR paradox for variables with a continuous spectrum has previously been carried out [6,7], which when combined with our analysis, forms the basis of a realizable experiment to teleport the complete quantum state of a single mode of the electromagnetic field. Note that up until now, all experimental proposals for teleportation have involved dichotomic variables in SU(2) [2,8-11], with optical schemes accomplishing the Bell-operator measurement with low efficiency. Indeed, the recent report of teleportation via parametric down conversion [12] succeeds only a posteriori with rare post-selected detection events. By contrast, our scheme employs linear elements corresponding to operations in SU(1,1) [13] for Bell-state detection and thus should operate at near unit absolute efficiency, enabling a priori teleportation as originally envisionaged in Ref. [2], As shown schematically in Fig. 1, an unknown input state described by the Wigner function Wm(a) is to be teleported to a remote station, with the teleported (output) state denoted by Wout(a). In analogy with the previously proposed scheme for teleportation of the state of a spin-^ particle, Alice (at the sending station) and Bob (at the receiving terminal) have previously arranged to share an entangled state which is sent along paths 1 and 2. Within the context of our scheme in SU(1,1), the entangled state distributed to Alice and Bob is described by the Wigner function WEPR( {0,1}'. The
function is guaranteed to be a two-to-one function with some periodicity r € {0,1} i.e. f(x) = f(y) iff y = x + r (this is a bit by bit addition mod2) for all x, y € {0,1}'. The goal is to determine r. The classical (probabilistic) algorithm which gives r with some fixed probability requires exponential number of /-evaluations. The quantum algorithm computes / only m times where m is a number which is of the order of I. The algorithm proceeds as follows. • Start with the first and the second register (both with I qubits) in state | 0) and apply the Hadamard transform to the first register to get
\x)\°)-
E
(21)
x£{0,l}'
• Evaluate / . This gives
E
I *>!/(*)>•
(22)
(|A> + | * + r))|/(fc)>
(23)
a;e{0,l} 1
• Measure the second register to get
for some k. • Apply the Hadamard transform to the first register to get
E !/(*)>
(24)
I/6{0,1}'
• Measure the first register. Prom the last equation above it follows that if r • y = 1 then (1 + (—l) ry ) = 0 and the probability amplitude of state \y) is zero. Thus the measurement must give y such that y • r = 0. • Repeat the above to find enough different j/j's so that r can be determined by solving the system of linear equations y\ • r = 0 , . . . ym • r = 0. The quantum factoring algorithm [7] was inspired by Simon's quantum periodicity estimation. Shor's quantum factoring of an integer n is based on calculating the period of the function Fn{x) = ax mod n for a randomly selected integer a between 0 and n. It turns out that for increasing powers of o, the remainders form a repeating sequence with a period which we denote r. Once r is known the factors of n are obtained by calculating the greatest common divisor of n and arl2 ± 1. Suppose we want to factor 15 using this method. Let a — 11. For increasing x the function l l x mod 15 forms a repeating sequence 1,11,1,11,1,11, The period is r = 2, and aTl2 mod 15 = 11. Then we take the greatest common divisor of 10 and 15, and of 12 and 15 which gives us respectively 5 and 3, the two factors of 15. Classically calculating r is at least as difficult as trying to factor n; the execution time of calculations grows exponentially with the number of digits in n. Quantum computers can find r in time which grows only as a cubic function of the number of digits in n. The structure of Shor's algorithm is very similar to Simon's periodicity estimations but unlike its predecessors Shor's algorithm does not involve any "black boxes" (or "oracles"). To estimate the period r we prepare two quantum registers; the first register, with I qubits, in the equally weighted superposition of all numbers it can contain, and the second register in state zero. Then we perform an arithmetical operation that takes advantage of quantum parallelism by computing the function Fn(x) for each number x in the superposition. The values of Fn(x) are placed in the second register so that after the computation the two registers become entangled:
EI^IO^EI^I^)) •
(25)
Now we perform a measurement on the second register. We measure each qubit and obtain either "0" or " 1 " . This measurement yields value Fn{k) (in binary notation) for some randomly selected k. The state of the first register right after the measurement, due to the periodicity of Fn(x), is a coherent superposition of all states | x) such that x = k, k + r, k + 2r,..., i.e. all x for which Fn(x) - Fn(k). The periodicity in the probability amplitudes in the first register cannot be simply measured because the offset i.e. the value k is randomly selected by the measurement. However the state of the first register can be subsequently transformed via a unitary operation which effectively removes the offset and modifies the period in the probability amplitudes from r to a multiple of 2l/r. This operation is known as the quantum Fourier transform (QFT) and can be written as 2'-l
QFT : | x) H-> 2~1'2 Y^ exp(2nixy/2l)
| y).
(26)
j/=0
After Q F T the first register is ready for the final measurement which yields with high probability an integer which is the best whole approximation of a multiple of 2l/r i.e. x = m2l/r for some integer m. We know the measured value x and the size of the register I hence if m and r are coprime we can determine r by canceling x/2l down to an irreducible fraction and taking its denominator. Since the probability that m and r are coprime is sufficiently large (greater than 1/logr for large r) this gives an efficient randomised algorithm for determination of r. A more detailed description of Shor's algorithm can be found in [7, 19]. Perhaps the most important recent development in quantum algorithms is Grover's 'quantum database search' [13]. Suppose we are given (as an oracle) a function fk which maps {0,1}' to {0,1} such that fk{x) = 5xk for some k. Our task is to find k. Thus in a set of numbers from 0 to 2l — 1 one element has been "tagged" and by evaluating ff. we have to find which one. To find k with probability of 50% any classical algorithm, be it deterministic or randomised, will need to evaluate fk a minimum of 2 ' - 1 times. In contrast, Grover's quantum algorithm needs only 0 ( 2 ' / 2 ) evaluations. Again, the first register is prepared in state | 0) and the second one (containing only one qubit) in state 11). The the sequence of operations KUfk H Uf0, where H is the Hadamard transform performed both on the first and the second register and Ufk is the quantum evaluation of fk, is repeated roughly 2 ( / 2 times and the first register is measured. The result of the measurement is, with probability greater than half, the sought after value k. Although the speed-up the data base search remains computationally hard even for quantum computers (the speed-up is only by the square root factor), but it is truly remarkable that such a difficult problem as searching an unstructured data base can be improved at all. It seems that Grover's algorithm probes the very limits of power of quantum computation (see, for example, discussion by Bennett et al. [20]).
4
Quantum Networks
Our description of quantum algorithms, as presented in the previous section, is of little help to those who would like to make a practical use of the awesome computational power of quantum devices. Clearly we cannot just assume that any unitary transformation U representing a mathematical operation, e.g. a function evaluation
X
X
or the quantum Fourier transform (Eq. 26), may be efficiently implemented. We have to show how to construct U using some finite basic set of transformations. In classical computation any Boolean function can be constructed as a Boolean network composed of elementary logic gates. Extending this idea to the quantum domain we will now express quantum algorithms in terms of quantum computational
networks. A quantum network, introduced by Deutsch [21] in 1989, is a quantum computing device consisting of quantum logic gates whose computational steps are synchronised in time. The outputs of some of the gates are connected by wires to the inputs of others. Quantum gates are the active components of quantum networks - in an n-bit quantum gate, n qubits undergo a coherent interaction. Wires are the passive components which allow to carry quantum states from one computational step to another. However, the distinction between "active" and "passive" components can only be made relative to a given physical and technological implementation. Different technologies would lead us to draw the line differently. In most current quantum technologies there are no physical "wires" to move qubits into adjacent positions so that they can undergo a gate-type interaction - a "wire" is just a convenient symbol which acts as a time-separator between two subsequent computational steps. So how many quantum gates do we need? In fact only one and almost any two-qubit gate can act as a universal quantum gate [22, 23]! Of course, since there is a continuum of possible unitary transformations quantum networks in general will only approximate these transformations, but any desired degree of approximation can be obtained using sufficiently long networks. Any quantum algorithm can be represented as a family of networks, for example, quantum addition is represented by a family of quantum adders, each adder in the family acts on a different number of input qubits. The computational complexity of a family of networks {Qi} may now be defined in terms of the size of Qi i.e. the number of gates involved in Qi. An algorithm will be called efficient if it has a (uniform) family of polynomial-size networks i.e. a family {Q;} such that the size of Qi grows only polynomially with I. "Uniform" means that it is easy to construct the whole family of networks e.g. the network has a pattern which allows to build Qi+i as a simple extension of Qi (c.f. the QFT network below). Let us introduce some elementary gates. The Hadamard gate implements the Hadamard transform (Eq.2). It is the single qubit gate H performing the unitary transformation H
-V f2 U1
l "I
\x)
(-l)«|i) + | l - x )
H
(28)
where the diagram on the right provides a schematic representation of the gate H acting on a qubit in state | a;), with x = 0 , 1 . The conditional phase shift is the two-bit gate B() defined as
B(0) =
/ 1 0 0 0 0 1 0 0 0 0 1 0
\x)
\
ixy<j> |
x)\y)
•
(29)
\y)
\ 0 0 0 e^ J
The matrix is written in the basis {| 0) 10), 10) 11), 11) | 0), 11) 11)} (the diagram on the right shows the structure of the gate). Another important two-qubit gate is the quantum controlled-NOT (or XOR) operation defined as ( 1 0 0 0 1 0 C = 0 0 0
0 \ 0 1
Vo o i o J
\x)
\x)
\v)
\x®y)
(30) &
where x, y = 0 or 1 and © denotes XOR or addition modulo 2. The quantum controlled-NOT gate is not a universal gate but a universal quantum gate can be constructed by a combination of the controlled-NOT and simple unitary operations on a single qubit (for more details about the relevance and possible implementations of this gate see [24, 25]). Suppose we want to implement Shor's factoring algorithm, i.e. we want to build a dedicated quantum device to factor large integers, how shall we start? Firstly we notice that quantum factorisation contains two major operations: quantum exponentiation (computing ax mod n) followed by the quantum Fourier transform. Quantum exponentiation can be
\x)
\x)
\v) l*>
- &
l»>
I v)
\xy ® z)
|0)
TOFFOLI GATE
- & Q
SUM CARRY
\x®y) \xy)
QUANTUM ADDER
Figure 2: Diagrammatic representation of the controlled-controlled-NOT (Toffoli) gate and a simplified quantum adder. States | a ; ) , | y ) , and \z) belong to the computational basis x,y,z = 0 or 1 and both addition © and multiplication xy are performed modulo 2. The Toffoli gate is a very handy basic unit which features prominently in the network implementing elementary quantum arithmetic i.e. in quantum adders, multipliers etc. It can be decomposed and written as a quantum network of elementary two-qubit and one-qubit gates. A simplified quantum adder is a starting point for constructing more elaborate networks. decomposed into a sequence of squaring, ax=a2°xo-a2l^-...a2'~lx'-K
(31)
where XQ,XI ... are the binary digits of x. Squaring is achieved by multiplication and multiplication by a sequence of additions. Following this reduction procedure we end up with a quantum adder as a basic unit for the whole network. However, a quantum adder is different from a classical adder. Any unitary operation is reversible which is why quantum networks for basic arithmetic cannot be directly deduced from their classical Boolean counterparts (classical logic gates such as AND or OR are clearly irreversible - reading 1 at the output of the OR gate does not provide enough information to determine the input which could be either (0,1) or (1,0) or (1,1)). Quantum arithmetic must be build ab initio from the reversible logical components. A good starting point is a simplified quantum adder shown in Fig.2. Explicit constructions of more elaborate quantum networks leading to modular exponentiation have been described in detail by Vedral et al. [26] and Beckman et al. [27]. The second part of the Shor algorithm, i.e. the discrete quantum Fourier transform (Eq. 26), is much easier to implement. The QFT network will consist of only one-qubit and two-qubit gates, which are: the Hadamard gate H and the conditional phase shift B(0). In Fig.3 we show the network which performs the quantum Fourier transform for / = 4 (for more details on the implementations of QFT see [28]). A general case of / qubits requires a trivial extension of the network following the same sequence pattern of gates H and B . The QFT network operating on I qubits contains I Hadamard gates H and /(/ — l ) / 2 phase shifts B , in total 1(1 + l ) / 2 elementary gates. Thus the quantum Fourier transform can be performed in an efficient way, the network size grows only as a quadratic function of the size of the input.
5
Conditional quantum dynamics
In the past three years researchers from different walks of physics have proposed many technologies for quantum logic gates ranging from the cavity QED [29] via ions in linear traps [30, 31] to the NMR based bulk spin computation [32, 33]. Single qubit quantum gates are regarded as relatively easy to implement. For example, a typical quantum optical realisation uses atoms as qubits and controls their states with laser light pulses of carefully selected frequency, intensity and duration; any prescribed
|x0)
| 0) + e 2 ^ / 2 11)
H
| 0) + e2™*/22 11)
H l*2>
| o) + e 2 W 2 3 | x)
H
•—•
| 0) + e 2 " " / 2 " 11)
H |
\x3)
H B(TT) H B(7r/2)B(7r) H B(7r/4)B(7r/2)B(7r) H
Figure 3: The quantum Fourier transform (QFT) network operating on four qubits. If the input state represents number x — J^k ^kxk the output state of each qubit is of the form | 0) + e^* 11), where k = 27T/2* and k — 0 , 1 , 3 . . . . N.B. there are three different types of the B(0) gate in the network above: B(TT), B ( T T / 2 ) and
B(TT/4).
I1)]
|1> Y
hr,M)JV
V Hi —
hio\oz
H2 = h,uJioz '
Figure 4: The control qubit of resonant frequency wi interacts via V with the target qubit of resonant frequency W2- Due to the interaction the two resonant frequencies are modified and the combined system of the two qubits has four different resonant frequencies wi ± fi and u>2 ± ^- A 7r-pulse at frequency LO2 + H causes the transition | 0) «-> 11) in the second qubit only if the first qubit is in state 11). This is one possible realisation of the quantum controlled-NOT gate. superposition of two selected atomic states can be prepared this way. Two-qubit gates are much more difficult to build. In order to implement two-qubit quantum logic gates it is sufficient, from the experimental point of view, to induce a conditional dynamics of physical bits, i.e. to perform a unitary transformation on one physical subsystem conditioned upon the quantum state of another subsystem, U = | 0) (0 | ® UQ + 11) (11 ® Ui + ... + | k) (k | ® Uk,
(32)
where the projectors refer to quantum states of the control subsystem and the unitary operations Ui are performed on the target subsystem [24]. The simplest non-trivial operation of this sort is probably a conditional phase shift such as TZ(<j>) which we used to implement the quantum Fourier transform and the quantum controlled-NOT (or XOR) gate. Let us illustrate the notion of the conditional quantum dynamics with a simple example (see Fig.4). Consider two qubits, e.g. two spins, atoms, single-electron quantum dots, which are coupled via Oz'crz' interaction (e.g. a dipole-dipole interaction). The first qubit with the resonant frequency a>i
77 will act as the control qubit and the second one, with the resonant frequency ui2, as the target qubit. Due to the coupling V the resonant frequency for transitions between the states | 0) and 11) of one qubit depends on the neighbour's state. The resonant frequency for the first qubit becomes w\ ± Q depending on whether the second qubit is in state | 0) or 11). Similarly the second qubit's resonant frequency becomes u>2 ± fi, depending on the state of the first qubit. Thus a 7r-pulse at frequency U2 + H causes the transition | 0) ++ 11) in the second qubit only if the first qubit is in 11) state. This way we can implement the quantum controlled-NOT gate. Thus in principle we know how to build a quantum computer; we can start with simple quantum logic gates and try to integrate them together into quantum networks. However if we keep on putting quantum gates together into networks we will quickly run into some serious practical problems. The more interacting qubits are involved the harder it tends to be to engineer the interaction that would display the quantum interference. Apart from the technical difficulties of working at single-atom and single-photon scales, one of the most important problems is that of preventing the surrounding environment from being affected by the interactions that generate quantum superpositions. The more components the more likely it is that quantum computation will spread outside the computational unit and will irreversibly dissipate useful information to the environment. This degrading effect of the computer-environment interaction on the computer is generally known as decoherence [34]. Thus the race is to engineer sub-microscopic systems in which qubits interact only with themselves but not with the environment.
6
Stability of Quantum Computation
When we analyse physically realisable computations we have to consider errors which are due to the computer-environment coupling and from the computational complexity point of view we need to assess how these errors scale with the input size /. If the probability of an error in a single run, e(Z), grows exponentially with I, i.e. if e(l) = 1 — Aexp(-al), where A and a are positive constants, then the randomised algorithm cannot technically be regarded as efficient any more regardless of how weak the coupling to the environment may be. Unfortunately the computer-environment interaction leads to just such an unwelcome exponential increase of the error rate with the input size [35]. It is clear that for quantum computation of any reasonable length to ever be physically feasible it will be necessary to incorporate some efficiently realisable stabilisation scheme to combat the effects of decoherence. Deutsch discussed this problem during the Rank Prize Funds Mini-Symposium on Quantum Communication and Cryptography, Broadway, England in 1993 and proposed 'recoherence' based on a symmetrisation procedure (for details see [36]). The basic idea is as follows. Suppose we have a quantum system, we prepare it in some initial state | \P;) and we want to implement a prescribed unitary evolution | $(£)) or just preserve | ;) for some period of time t. Now, suppose that instead of a single system we can prepare R copies of | $j) and subsequently we can project the state of the combined system into the symmetric subspace i.e. the subspace containing all states which are invariant under any permutation of the sub-systems. The claim is that frequent projections into the symmetric subspace will reduce errors induced by the environment. The intuition behind this concept is based on the observation that a prescribed error-free storage or evolution of the R independent copies starts in the symmetric sub-space and should remain in that sub-space. Therefore, since the error-free component of any state always lies in the symmetric subspace, upon successful projection it will be unchanged and part of the error will have been removed. Note however that the projected state is generally not error-free since the symmetric subspace contains states which are not of the simple product form | \P) | \ P ) . . . | $ ) . Nevertheless it has been shown that the error probability will be suppressed by a factor of 1/R [36]. More recently projections on symmetric subspaces were replaced by more complicated projections on carefully selected subspaces. These clever projections, proposed by Shor [37], Calderbank and Shor [38], Steane [39] and others [40, 41, 42, 43, 44], are constructed on the basis of classical errorcorrecting methods but represent intrinsically new quantum error-correction and stabilisation schemes;
they are the subject of much current study. There are also other approaches to recoherence [45, 46, 47] but here we illustrate the main idea with the simplest three-qubit quantum error-correcting code. Within a simplified model of the computer-environment interaction, known as decoherence [34], it is assumed that the register in the computer and the environment undergo the following unitary evolution \i)\R)-+\i)\Ri(t)), (33) where | i) is the state from the computational basis and | R) is the initial state of the environment. States | Ri(t)) are normalised but not necessarily orthogonal to each other. Now, consider the following initial state of the computer and the reservoir
|¥(0)) = 5 > | * > ® | « > -
(34)
i
The unitary evolution of the composed system results in an entangled computer-reservoir state which can be written as \9(t)) = ^2a\i)®\Ri(t)), (35) i
where, in general, {Ri \ Rj) ^ 0 for i / j . The elements of the density matrix evolve as Pij(0)
= a c* —+ Pij(i)
= a c* (Ri(t)\ Rj(t)).
(36)
In this popular model of decoherence [34] the environment effectively acts as a measuring apparatus; in time the reservoir states {| Ri)} become more and more orthogonal to each other whilst the coefficients {c;} remain unchanged. Consequently, after some period of time known as the decoherence time, the off-diagonal elements p^ disappear due to the (Ri(t) | Rj(t)) factors. If the computing device contains I qubits the typical decoherence time for the computer will be of the order ta/l, where tj is a decoherence time of a single qubit. Clearly decoherence puts an upper bound on the length of any feasible quantum computation. For if the elementary computational step takes time r then the requirement that a coherent computation of K steps involving I qubits be completed within the decoherence time of the computer can be written as TK
< Ull.
(37)
In the case of Shor's algorithm for factorisation of an /-bit number we may take K — I3 and (37) provides an upper bound
l) — Co | 0) + c\ | 1) and we know that any single qubit which is stored in a register undergoes a decoherence type entanglement with an environment (Eq. 36); in particular (co 10} + ci 11» | R) —> co 10) | Ro(t)) + a 11) |
Ri(t)).
(39)
To see how the state of the qubit is affected by the environment, we calculate the fidelity of the decohered state at time t with respect to the initial state | <j>) F{t) = {4>\p{t)\4>) ,
(40)
where p(t) is given by Eq. (36). It follows that F W = |co| 4 + |c 1 | 4 + 2|c 0 | 2 |c 1 | 2 Re[( J Ro(t)|i?iW)]-
(41)
The expression above depends on the initial state | ) and clearly indicates that some states are more vulnerable to decoherence than others. In order to get rid of this dependence we consider the average fidelity, calculated under the assumption that any initial state | )\o)
I a?) •
(62)
For encoded basis states of a single qubit | Co) and | C\) this implies [47] A* | Co) | 0)
—>
| Co) | a t ) ,
(63)
A*|Ci)|0)
—•
\Ci)\ak),
(64)
where Ak denotes all the possible types of independent errors affecting at most one of the qubits. The above requirement leads to the following unitarity conditions
|0>
[) A=sin 2
10)
7
-?
1 \/2
|1>
£(I°>-|1»
0) + | 1)) (2) • 00 + 00
(and extends by linearity to states of the form a | 0) + /311)). Here, we have ignored the el 2 phase shift in the reflected beam, which is irrelevant because the interference pattern depends only on the difference between the phase shifts in different arms of the interferometer. The phase shifters in the two paths can be tuned to effect any prescribed relative phase shift <j> — 4>o ~ 4>i and to direct the particle with probabilities cos 2 ( | J and sin 2 ( | J respectively to detectors "0" and " 1 " . The roles of the three key ingredients in this experiment are clear. The first beam splitter prepares a superposition of possible paths, the phase shifters modify quantum phases in different paths and the second beam-splitter combines all the paths together. As we shall see in the following sections, quantum algorithms follow this interferometry paradigm: a superposition of computational paths is prepared by the Hadamard (or the Fourier) transform, followed by a quantum function evaluation which effectively introduces phase shifts into different computational paths, followed by the Hadamard or the Fourier transform which acts somewhat in reverse to the first Hadamard/Fourier transform and combines the computational paths together. To see this, let us start by rephrasing Mach-Zehnder interferometry in terms of quantum networks.
2
Quantum gates &; networks
In order to avoid references to specific technological choices (hardware), let us now describe our MachZehnder interference experiment in more abstract terms. It is convenient to view this experiment as a quantum network with three quantum logic gates (elementary unitary transformations) operating on a qubit (a generic two-state system with a prescribed computational basis {| 0 ) , 11)}). The beamsplitters will be now called the Hadamard gates and the phase shifters the phase shift gates (see Fig. 2). The Hadamard gate is the single qubit gate H performing the unitary transformation known as the
H
H
Figure 2: A quantum network composed of three single qubit gates. This network provides a hardwareindependent description of any single-particle interference, including Mach-Zehnder interferometry.
10)
H
H
Measurement
U
!>
!>
Figure 3: Phase factors can be introduced into different computational paths via the controlled-?/ operations. The controlled-^/ means that the form of U depends on the logical value of the control qubit (the upper qubit). Here, we apply the identity transformation to the auxiliary (lower) qubits (i.e. do nothing) when the control qubit is in state | 0) and apply a prescribed U when the control qubit is in state 11). The auxiliary or the target qubit is initially prepared in state | ip) which is one of the eigenstates of U. Hadamard transform given by (Eq. 2) H
1 1
J_ 72
1 -1
\x)
H
| 0 ) + ( - 1 ) * 11)
(3)
The matrix is written in the basis { | 0 ) , | 1 ) } and the diagram on the right provides a schematic representation of the gate H acting on a qubit in state | x), with x — 0 , 1 . Using the same notation we define the phase shift gate as a single qubit gate such that | 0) >->• | 0) and 11) i-> e l * 11), 1 0
( e
\x)
\x)
(4)
Let us explain now how the phase shift 4> can be "computed" with the help of an auxiliary qubit (or a set of qubits) in a prescribed state | %l>) and some controlled-!/ transformation where U | V>) = e"^ \ tp) (see Fig. 3). Here the controlled-?/ is a transformation involving two qubits, where the form of U applied to the auxiliary or target qubit depends on the logical value of the control qubit. For example, we can apply the identity transformation to the auxiliary qubits (i.e. do nothing) when the control qubit is in state 10) and apply a prescribed U when the control qubit is in state 11). In our example shown in Fig. 3, we obtain the following sequence of transformations on the two qubits
|0>|^> A ^ ( | 0 ) + |1))|V)
c-U
^(|0) + e*|l»|V) e«*>(cosf | 0) + i sin f 11» | V) •
(5)
We note that the state of the auxiliary register |«/>), being an eigenstate of U, is not altered along
this network, but its eigenvalue ei(* is "kicked back" in front of the 11) component in the first qubit. The sequence (5) is equivalent to the steps of the Mach-Zehnder interferometer (1) and, as was shown in [2], the kernel of most known quantum algorithms.
3
The first quantum algorithm
Since quantum phases in interferometers can be introduced by some controlled-C/ operations, it is natural to ask whether effecting these operations can be described as an interesting computational problem. Suppose an experimentalist, Alice, who runs the Mach-Zehnder interferometer delegates the control of the phase shifters to her colleague, Bob. Bob is allowed to set up any value tf> = 4>o — = 0,n. It is best represented as the quantum network shown in Fig. 4, where the middle operation is the /-controlled-NOT, which can be defined as: \x)\y)i--^N\x)\y®f{x))
.
(6)
The initial state of the qubits in the quantum network is | 0) (| 0) — 11)) (apart from a normalization factor, which will be omitted in the following). After the first Hadamard transform, the state of the two qubits has the form (| 0) + 11))(| 0) - 11)). To determine the effect of the /-controlled-NOT on
|0>
H
H
I o> — I x>
Uf
Measurement
|0)-|1)
Figure 4: Quantum network which implements Deutsch's algorithm. The middle gate is the / controlled-NOT which evaluates one of the four functions / : {0,1} \-¥ {0,1}. If the first qubit is measured to be | 0), then the function is constant, and if 11), the function is balanced. this state, first note that, for each x € {0,1},
| x) (| o> - 11)) f~-^N | x) (| o e f{x)) - | i e f{x))) = (-i)' | x) (| o> -11))
(7)
Therefore, the state after the /-controlled-NOT is ((_l)/(o)|0)
+
(
_1)/(i)|1>)(|0)_|1)).
(8)
That is, for each x, the | x) term acquires a phase factor of ( — l ) ^ x \ which corresponds to the eigenvalue of the state of the auxiliary qubit under the action of the operator that sends | y) to \y®f(x)). This state can also be written as (_l)/(o)(|o) +
(_i)/(o)e/(i)|1))(|0)_|1)))
(9)
which, after applying the second Hadamard transform to the first qubit, becomes
(-i)/(0)|/(o)e/(i))(|o)-|i».
(10)
Therefore, the first qubit is finally in state | 0) if the function / is constant and in state 11) if the function is balanced, and a measurement of this qubit distinguishes these cases with certainty. The Mach-Zehnder interferometer with phases o and 4>\ each set to either 0 or 7r can be regarded as an implementation of the above algorithm. In this case, o and 4>\ respectively encode /(0) and / ( l ) (with 7r representing 1), and a single photon can query both phase shifters (i.e. /(0) and / ( l ) ) in superposition. More recently, this algorithm (Fig. 4) has been implemented using a very different quantum physical technology, nuclear magnetic resonance [4, 5]. More general algorithms may operate not just on single qubits, as in Deutsch's case, but on sets of qubits or 'registers'. The second qubit becomes an auxiliary register | tp) prepared in a superposition of basis states, each weighted by a different phase factor, 2iriy/2"
l
\v)-
(11)
v=o In general, the middle gate which produces the phase shift is some controlled function evaluation. A controlled function evaluation operates on its second input, the 'target', according to the state of the first input, the 'control'. A controlled function / applied to a control state | a;), and a target state | ip) gives
|*>|V>—Ha:)l^+ /(*)>•
(12)
where the addition is mod 2 m . Hence for the register in state (11) 2
m
-l
I x) J2
2
e~2niv/2m
I V) —* e2irif^l2m
m
-l
| x) J2
y=0
e-2^y+f^'2m
\ y + /(*)) = e2™**)/2™ | x) | }.
V=0
(13) Effectively a phase shift proportional to the value of f(x) is produced on the first input. We will now see how phase estimation on registers may be carried out by networks consisting of only two types of quantum gates: the Hadamard gate H and the conditional phase shift R(0). The conditional phase shift is the two-qubit gate R(0) defined as
B
R
,^
1 0 0 0 , 0 1 0 0
W = I o o i 0
0 0
\ ) eixv* \x)\y).
o
(14)
\y)
e^ /
The matrix is written in the basis {| 0) | 0 ) , | 0) 11), 11) | 0 ) , 11) 11)}, (the diagram on the right shows the structure of the gate). For some of the known quantum algorithms, when working with registers, the Hadamard transformation, corresponding to the beamsplitters in the interferometer, is generalised to a quantum Fourier transform.
4
Quantum Fourier transform and computing phase shifts
The discrete Fourier transform is a unitary transformation of a s-dimensional vector (/(0), / ( l ) , / ( 2 ) , . . . , f(s - 1)) -> (/(0), / ( l ) , 7 ( 2 ) , . . . , f(s - 1))
(15)
denned by: 1
s_1
f(y) = —J2e2™y/sf(x), V S
(16)
x=0
where f(x) and f(y) are in general complex numbers. In the following, we assume that s is a power of 2, i.e., s = 2" for some n; this is a natural choice when binary coding is used. The quantum version of the discrete Fourier transform (QFT) is a unitary transformation which can be written in a chosen computational basis {|0), | 1 ) , . . . , |2 n — 1)} as, 1 s_1 \x) >—> -T= Y, exp{2nixy/s)
\y).
(17)
More generally, the QFT effects the discrete Fourier transform of the input amplitudes. If QFT : £ / ( * ) | z > — > £ / ( » ) ! » > > x
(18)
y
then the coefficients f(y) are the discrete Fourier transforms of the
f(x)'s.
n
A given phase <j>x — 27ra;/2 can be encoded by a QFT. In this process the information about (j>x is distributed between states of a register. Let x be represented in binary as xo--.xn-i G {0, l } n , Xi< an where x = 52^=0 ^" ( d similarly for y). An important observation is that the QFT of x, 53«=o exp(2TTixy/s) \y), is unentangled, and can in fact be factorised as (| 0) + e * - 11»(| 0) + e i2 *- 11)) • • • (| 0) + e42*"1*- 11)) .
(19)
\xo)
l*l>
H
I 0)
»
*
+
e27Tix/2
I ^
I 0) + e2vria:/22 | ^
# •—•
l*2>
J g)
H
l*3>
•—•
ff
I Q)
+
+
e2™/2
e2^«/2
3
4
I !)
J ^
H R(TT) H R(7r/2)R(7r) H R(7r/4)R(7r/2)R(7r) H
Figure 5: The quantum Fourier transform (QFT) network operating on four qubits. If the input state represents number x = J^k ^kxk the output state of each qubit is of the form | 0) + e'2" ^ 11), where <j>x = 2nx/2n and k = 0,1,2 .. .n - 1. N.B. there are three different types of the R(((>) gate in the network above: R(ir), R(n/2) and R(n/4). The size of the rotation is indicated by the distance between the 'wires'. The network for performing the QFT is shown in Fig. 5. The input qubits are initially in some state I x) = I xo) \xi)\ X2) I X3) where xoXix2x3 is the binary representation of a;, that is, x = Y^-o x^' • As the number of qubits becomes large, the rotations R(n/2n) will require exponential precision, which is impractical. Fortunately, the algorithm will work even if we omit the small rotations, [6, 7]. The general case of n qubits requires a simple extension of the network following the same pattern of H and R gates. States of the form (19) are produced by function evaluation in a quantum computer. Suppose that U is any unitary transformation on m qubits and | tp) is an eigenvector of U with eigenvalue e**. The scenario is that we do not explicitly know U or | ip) or e'*, but instead are given devices that perform controlled-U, controlled-f/ 2 , controlled-U2 and so on until we reach controlled-U2" . Also, assume that we are given a single preparation of the state \ip). From this, our goal is to obtain an n-bit estimator of d>. In a quantum algorithm a quantum state of the form | 0 > + e i2"
_
l » ( | 0 ) + e t2
n-
|l)).--(|0> + e * | l ) )
(20)
is created by applying the network of Fig. 6. Then, in the special case where
) can be obtained by just applying the inverse of the QFT (which is the network of Fig. 5 in the backwards direction and with the qubits in reverse order). If x is an n-bit number this will produce the state | xo • • • xn-\) exactly (and hence the exact value 4>). However, <j> is not in general a fraction of a power of two (and may not even be a rational number). For such a <j> = 2-KUJ, it turns out that applying the inverse of the QFT produces the best n-bit approximation of u) with probability at least 4/TT2 « 0.41 [2]. The probability of obtaining the best 1 estimate can be made 1 — 5 for any 6, 0 < S < 1, by creating the state in equation (20) but with n + 0(log(l/, since, for example, we might be able to to obtain as close an estimate with higher probability. See [8] for details.
|0) + |1)
I 0) + ei2 * 11)
|0) + |1)
| 0) + e i21 * 11)
|0) + |1)
|0> + e i 2 ° ^ | l )
I*)
u2
u2
U2
l*>
Figure 6: The network which computes phase shifts in Shor's algorithms; it also implements the modular exponentiation function via repeated squarings.
5
Examples
We will now illustrate the general framework described in the preceding section by showing how some of the most important quantum algorithms can be viewed in this light. We start with Shor's quantum algorithm for efficient factorisation (for a comprehensive discussion of quantum factoring see [9, 10, 2]).
5.1
Quantum Factoring
Shor's quantum factoring of an integer TV is based on calculating the period of the function f(x) = ax mod TV for a randomly selected integer a between 1 and TV. For any positive integer y, we define y mod TV to be the remainder (between 0 and TV- 1) when we divide y by TV. More generally, y mod TV is the unique positive integer y between 0 and TV - 1 such that TV evenly divides y-y. For example, 2 mod 35 = 2, 107 mod 35 = 2, and - 3 mod 35 = 32. We can test if a is relatively prime to TV using the Euclidean algorithm. If it is not, we can compute the greatest common divisor of a and TV using the extended Euclidean algorithm. This will factor TV into two factors TVi and TV2 (this is called splitting TV). We can then test if TVi and TV2 are powers of primes, and otherwise proceed to split them if they are composite. We will require at most log2 (TV) splittings before we factor TV into its prime factors. These techniques are summarised in [11]. It turns out that for increasing powers of a, the remainders form a repeating sequence with a period r. We can also call r the order of a since ar — 1 mod TV. Once r is known, factors of TV are obtained by calculating the greatest common divisor of TV and aTl2 ± 1. Suppose we want to factor 35 using this method. Let a = 4. For increasing x the function 4X mod 35 forms a repeating sequence 4,16,32, 29, 9,1,4,16, 2 9 , 3 2 , 9 , 1 , . . . . The period is r = 6, and arl2 mod 35 = 29. Then we take the greatest common divisor of 28 and 35, and of 30 and 35, which gives us 7 and 5, respectively, the two factors of 35. Classically, calculating r is at least as difficult as trying to factor TV; the execution time of the best currently-known algorithms grows exponentially with the number of digits in TV. Quantum computers can find r very efficiently. Consider the unitary transformation Ua that maps | x) to | ax mod TV). Such a transformation is realised by simply implementing the reversible classical network for multiplication by a modulo TV using quantum gates. The transformation Ua, like the element a, has order r, that is, UTa = 7, the identity operator. Such an operator has eigenvalues of the form e~^~ for k = 0,1, 2 , . . . , r — 1. In order to formulate Shor's algorithm in terms of phase estimation let us apply the construction from
the last section taking r-l
| ip) = "^2 e
-
^ I aj mod N) .
(21)
j=0
Note that \ip) is an eigenvector of Ua with eigenvalue e2nt(~K Also, for any j , it is possible to implement efficiently a controlled-t/, 2 ' gate by a sequence of squaring (since U%' = Ua2i )• Thus, using the state | ip) and the implementation of controlled-!/ 2 ' gates, we can directly apply the method of the last section to efficiently obtain an estimator of - . The problem with the above method is that we are aware of no straightforward efficient method to prepare state | ip), however, let us notice that almost any state | ipf.) of the form r-l
\'>Pk)=J2e-^\aj
mod N) ,
(22)
j=o
where k is from { 0 , . . . , r - 1} would also do the job. For each k G { 0 , 1 , . . . , r - 1}, the eigenvalue of state | ipk) is e 2 " ^ " ' . We can again use the technique from the last section to efficiently determine and if k and r are coprime then this yields 2 r. Now the key observation is that r
(23)
|l) = E l ^ > ' and 11) is an easy state to prepare.
If we substituted 11) in place of | ip) in the last section then effectively we would be estimating one of the r, randomly chosen, eigenvalues e2w'^\ This demonstrates that Shor's algorithm, in effect, estimates the eigenvalue corresponding to an eigenstate of the operation Ua that maps | x) t o | ax mod N). A classical procedure - the continued fractions algorithm - can be employed to estimate r from these results. The value of r is then used to factorise the integer.
5.2
Finding hidden subgroups
A number of algorithms can be generalised in terms of group theory as examples of finding hidden subgroups. For any g S G, the coset gK, of the subgroup K is defined as {gK\g G G}. Say we have a function / which maps a group G to a set X, and / is constant on each coset of the subgroup K, and distinct on each coset, as illustrated in Figure 7. In other words, f(x) — f(y) if and only if x — y is an element of K. In Deutsch's case, G = {0,1} with addition mod 2 as the group operation, and X is also {0,1}. There are two possible subgroups K: | 0), and G itself. We are given a black-box Uf for computing / \x)\y)->\x)\y®
f{x)).
There are two cosets of the subgroup {0}: {0} and {1}. If the function is defined to be constant and distinct on each coset, it must be balanced. On the other hand, there is only one coset of the 2
If the estimate y/2m of k/r satisfies
I 2 m ~ r I 2JV2 ' then there is a unique rational of the form ^ with 0 < b < N satisfying \ y a\ \2™ ~~b~\
