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John by contributed s toe n reyrve h progress the review briefly I and one, ct hgnlqatmsae.Asml proof simple A states. quantum thogonal udhv arsoial distinguishable macroscopically have ould f biu htte i o eev obe to deserve not did they that obvious tcnieal neetadta finding that and interest considerable it rte najro nopeesbeto incomprehensible jargon a in written lwn er[].Ohrojcin were objections Other [7]). year llowing dcinwsipiil sdb Stephen by used implicitly was rdiction nn hoe,md ulcfrtefirst the for public made theorem, oning onain fPhysics of Foundations Aspruia omnctrbased communicator superluminal —A t’s unu hoy tasrsta no that asserts It theory. quantum n ddt eetHretsppr His paper. Herbert’s reject to nded neetGopi loihsand Algorithms in Group Interest l prcudntb orc,bcueit because correct, be not could aper FLASH nbet ie eas fspace of because cite, to unable 1] lotctyasmsthat assumes tacitly also [10], ae,koigperfectly knowing paper, se et be to me asked amiliar. udnot ould Anyway, the no-cloning theorem was no justification for rejecting Herbert’s paper. His manuscript mentioned exact duplication as a theoretical ideal, but it was clear that what was actually produced by the laser gain tube was a messy state, where many different outputs were entangled with the laser final state. The no-cloning theorem and its practical implications for quantum cryptography led to an enormous wave of interest, as I had predicted. Every month one or more papers on this subject are put in the quant-ph electronic archive or appear in other media. Typical questions that are discussed are how to achieve optimal, if not perfect cloning [11], a question which led to the important notion of quantum disentanglement [12]; the no-broadcasting theorem [13], which is a generalization of no-cloning to general impure density matrices; M → N probabilistic cloning [14, 15]; rigorous and stronger theorems [16, 17, 18]; and in particular the relation of optimal cloning to the impossibility of signaling [19, 20, 21] and bounds on state estimation that are imposed by the no signaling condition [22]. The problem was finally laid to rest by a fundamental result: completely positive maps, the only ones that can be realized by quantum mechanical systems, cannot increase distinguishability [23]. There is no doubt that the term “superluminal” was one of the causes this subject became so attractive. Who would not be happy to beat the relativistic limit on the speed of transmission of information? Actually, superluminal group velocities have been observed in barrier tunneling in condensed matter [24, 25]. However, special relativity does not forbid the group velocity to exceed c. It is the front velocity of a wave packet that is the relevant criterion for signal transmission, and the front velocity never exceeds c. Clearly, the term superluminal has no room in the present considerations. The never enters in them, nor does the . The issue simply is: given two observers with an entangled bipartite quantum system (or even numerous such systems: they are equivalent to a single, bigger, physical system) it is impossible, by means of local quantum operations (LQO), to transmit any information whatsoever from one observer to the other, without transmitting real material objects between them. This is what says [26], and a speculative question could be: what kinds of modifications of quantum mechanics would allow information transfer by means of LQOs. In the following, I’ll show that a nonlinear evolution of the ρ would give such a result. A similar conclusion was also reached by Svetlichny [27], who however only considered the special case of an initial maximally entangled pure state, and made further restrictive assumptions (and also used the term “superluminal” without ever using the theory of relativity). Consider our two familiar observers, Alice and Bob, who have a bipartite quantum system in a known state ρ. They perform LQOs that are mathematically represented by positive operator valued measures (POVMs) with elements Aj = 1l and Bµ = 1l respectively. The probability for the joint result jµ is P P Pjµ = TrATrB (ρ Aj ⊗ Bµ), (1) where the double trace is taken on the indices of Alice and of Bob. If Bob is not informed that Alice got result j, the probability that he gets µ is

Pjµ = Tr(ρB Bµ), (2) Xj where ρB = TrA(ρ) is Bob’s reduced density matrix. This result does not depend on Alice’s choice of a POVM. This is what quantum mechanics says, and all this is well known. If Bob is informed that Alice got result j, it follows from (1) that the probability that Bob has result µ is TrB(ρj Bµ), where

ρj = TrA(ρAj )/pj . (3)

Here the denominator pj = ν Pjν is the probability that Alice gets result j, so that ρj has unit trace as it should. P Therefore everything happens as if the state of Bob’s subsystem actually was ρj . Quantum mechanics does not claim that this is true, but also gives no way of showing that this realistic point of view is false. It’s just a matter of belief. Bernard d’Espagnat calls Bob’s reduced density matrix an “improper” mixture [28]. I also considered such situations in [4], where I distinguished pure states, mixtures, and “compounds” (a compound is a mixture that has a unique decomposition into pure states if additional

2 information is supplied). Here, I am slightly more general: Bob’s reduced density matrix can be split in a unique way into other density matrices (not only pure states) if Alice reveals to him which result she got. The question is whether Bob can do that without Alice’s help, and thereby find out which POVM she chose to perform. For this, one has to violate quantum mechanics in some way, such as cloning, as originally proposed. Here, it is essential to assume that cloning an improper mixture (a “compound”) means to clone each component separately, as if the state really was one of these components, with probability pj , and we merely ignored which one. Let us suppose that Bob can perform a trace-preserving nonlinear transformation

ρj → ρ˜j , (4) on the unknown “true” states of his subsystem. Then his reduced density matrix evolves as

ρB = pj ρj → pj ρ˜j . (5) Xj Xj

Note that the result is not determined by ρB alone, but the decomposition of ρB into a definite set of ρj is essential. In particular, the result depends on the choice made by Alice of a particular POVM, as Eq. (3) explicitly shows. If Alice chooses a different POVM, As = 1l, giving Bob states = TrA(ρAs)/ps with probability ps, Bob would obtain, after the hypotheticalP nonlinear evolution,

ρB = psρs → psρ˜s. (6) Xs Xs Let us call the right hand sides of the last two equations ρ′ and ρ′′, respectively. In the generic case, they are not equal to each other and they can be distinguished statistically. If enough copies are supplied to Bob, he would be able to know which POVM Alice chose. Needless to say, the assumption of a nonlinear evolution of ρ violates quantum mechanics. Contrary- wise, if ρ evolves linearly, then ρ′ = ρ′′ and no information can be transmitted in this way. Note that all the discussion is about the density matrix ρ. It is quite possible to have a nonlinear evolution of pure states which corresponds to a linear evolution of ρ. For example, the evolution

α 1 → (7) β 0 is nonlinear in terms of pure states, because

α 1 0 ≡ α + β , (8) β 0 1 1 and the right hand side would give (α + β) 0 if the evolution were linear. On the other hand, the same evolution expressed in terms of density matrices appears as a b 1 0 → . (9)  b∗ 1 − a 0 0 This is a linear operation, generated by a pair of Kraus matrices

1 0 0 1 and . (10) 0 0 0 0

In summary, Nick Herbert’s erroneous paper was a spark that generated immense progress. There also are many wrong papers that have been published in reputable journals, some of them by renowned scientists. Their bad influence may last for years. For these, I decline all responsibility. I was not the referee of these papers and I could not protect the good reputation of their authors.

I am grateful to Erika Andersson, Gilles Brassard, Dagmar Bruß, Chris Fuchs, GianCarlo Ghirardi, Nobu Imoto, Danny Terno, and Bill Wootters for many clarifying comments and for helping me to locate some of the references below. This work was supported by the Gerard Swope Fund and the Fund for Encouragement of Research.

3 References

[1] W. K. Wootters and W. H. Zurek, Nature 299 (1982) 802. [2] D. Dieks, Phys. Letters 92A (1982) 271. [3] A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1995) p. 279. [4] A. Peres, in Mathematical Foundations of Quantum Theory, ed. by A. R. Marlow (Academic Press, New York, 1978), p. 357. [5] N. Herbert, Found. Phys. 12 (1982) 1171. [6] R. Kipling, Just So Stories (MacMillan, London, 1902). [7] G. C. Ghirardi and T. Weber, Nuovo Cimento 78B (1983) 9. [8] R. J. Glauber, in New Techniques and Ideas in Quantum Measurement Theory, ed. by D. M. Green- berger, Ann. New York Acad. Sci. 480 (1986) 336.

[9] S. Wiesner, SIGACT News, 15 (1983) 78. [10] C. H. Bennett, G. Brassard, S. Breidbart, and S. Wiesner, in Advances in Cryptology (Proceedings of Crypto-82, Plenum, New York, 1983) p. 267. [11] D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Phys. Rev. A57 (1998) 2368. [12] D. R. Terno, Phys. Rev. A59 (1999) 3320. [13] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev. Lett. 76 (1996) 2818. [14] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 80 (1998) 4999. [15] N. J. Cerf and S. Iblisdir, Phys. Rev. A 62 (2000) 040301. [16] G. Lindblad, Lett. Math. Phys. 47 (1999) 189. [17] M. Koashi and N. Imoto, quant-ph/0101144. [18] R. Jozsa, quant-ph/0204153. [19] N. Gisin, Phys. Lett. 242A (1998) 1. [20] S. Ghosh, G. Kar, and A. Roy, Phys. Letters 261A (1999) 17. [21] D. Bruß, G. M. D’Ariano, C. Macchiavello, and M. F. Sacchi, Phys. Rev. A 62 (2000) 062302. [22] S. M. Barnett and E. Andersson, Phys. Rev. A 65 (2002) 044307. [23] C. A. Fuchs and J. van de Graaf, IEEE Trans. Info. Theory 45 (1999) 1216. [24] R. Y. Chiao and A. M. Steinberg, in Progress in Optics XXXVII , ed. by E. Wolf (Elsevier, Amster- dam, 1997); Physica Scripta T76 (1998) 61. [25] J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, Phys. Letters 245A (1998) 19. [26] A. Peres, Phys. Rev. A 61 (2000) 022117. [27] G. Svetlichny, Found. Phys. 28 (1998) 131. [28] B. d’Espagnat, Veiled Reality (Addison-Wesley, Reading, 1995) p. 105.

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