John von Neumann on quantum correlations

Transcription

1 John von Neumann on quantum correlations Miklós Rédei Department of History and Philosophy of Science Loránd Eötvös University P.O. Box 32, H-1518 Budapest 112, Hungary Abstract In an (unpublished) letter by von Neumann to Schrödinger (dated April 11, 1936) von Neumann replies to Schrödinger s two famous 1935 papers, in which the notion of entanglement between spatially separated quantum systems is introduced and the probabilistic correlations arising from entanglement is discussed from the perspective of a possible clash between quantum mechanics and the principle of physical locality. By quoting extensively from von Neumann s letter it will be seen that von Neumann position concerning such correlations is that they are unproblematic as long as (i) one can (at least in principle) assume that the correlations are explainable by common causes, or (ii) probabilities are interpreted subjectively. It will be argued that while a subjective interpretation of quantum probabilities is difficult to accept in a quantum context, a common cause type explanation of quantum correlations might be possible under a suitable specification of common cause. 1 The historical context In 1935 Einstein, B. Podolsky and N. Rosen published the famous EPR paper [1]. The paper s aim was to prove that one should consider the quantum mechanical description of physical reality incomplete provided that one accepts the principle of locality: that the physical state of a subsystem S 1 of a joint system (S 1 + S 2 ) cannot be changed instantaneously by performing a measurement on subsystem S 2 spatially separated from subsystem S 1. The discussions between Einstein, Rosen and Podolsky that led to the EPR paper were taking place in Einstein s office at the Institute for Advanced Study in Princeton in the spring of 1935 [7]. Von Neumann was Einstein s colleague at the Institute for Advanced Study and, given that the EPR paper concerned the completeness of quantum mechanics, in which von Neumann was very much interested (it is well known that he discussed completeness of quantum mechanics at length in his book [19], concluding, on the basis of the famous no-hidden variable proof, that quantum mechanics is complete), one would expect that Einstein and von Neumann exchanged ideas on this issue. Surprisingly, 1

2 this does not seem to be the case; to be more precise: the only record I know of that indicates an exchange between von Neuman and Einstein on this subject is von Neumann s unpublished letter to Schrödinger (April 11, 1936). This hand written letter is in the Archive for the History of Quantum Mechanics and it will be published in full in [21]. The letter starts with this sentence: Einstein has kindly shown me your letter as well as a copy of the Pr.Cambr.Phil.Soc. manuscript. I feel rather more overquoted than under-quoted and I feel that my merits in the subject are over-emphasized. (Von Neumann to Schrödinger, April 11, 1936), [20] So Einstein did talk to von Neumann about quantum mechanics after all and tried to raise von Neumann s interest in Schrödinger s paper: Probability relations between separated systems, Proceedings of the Cambridge Philosophical Society, 32 (1936) , [15]. This paper is the second of the two famous papers Schrödinger wrote in 1935 [14] and [15]. We know that Einstein and Schrödinger corresponded about the EPR paper in the summer of Schrödinger s above mentioned two papers were motivated by the EPR paper and by his correspondence with Einstein. Von Neumann s letter to Schrödinger (April 11, 1936) is a direct reply to Schrödinger s second paper [15]. 2 Summary of Schrödinger s treatment and interpretation of entanglement Schrödinger argues in both [14] and [15] that the presence of correlations between spatially separated quantum subsystems of a joint quantum system threatens the principle of locality and thereby might be in contradiction with the theory of relativity. Specifically, Schrödinger considers a composite quantum system described by the tensor product Hilbert space H 1 H 2, where H 1 and H 2 are assumed to be identical copies of an L 2 function space describing system S 1 and S 2, respectively. Schrödiner shows in [14] that any state vector Ψ(x, y) H 1 H 2 of the composite system can be written as Ψ(x, y) = a k g k (x) f k (y) g k H 1 f k H 2, a k IC (1) k with {g k } and {f k } being complete sets of orthogonal (unit) vectors in the respective spaces (not all a k necessarily nonzero). The decomposition (1) is called the biorthogonal decomposition and it is unique (up to relabelling of the elements g k and f k respectively). Vectors f k and g k can be viewed as eigenvectors (with eigenvalues λ F k, λ G k ) of some observables F and G of system S 1 and S 2, respectively. If we carry out a measurement of G on S 2 and find eigenvalue λ G n then... we have to assign to the first system the wave function g k (x). [15][p. 450]. From the perspective of system S 1, the state of the joint system (S 1 + S 2 ) given by Ψ(x, y) differs from its state given by g k because the 2

3 state of S 1 given by g k is a pure state on S 1, whereas the state given by vector Ψ(x, y) is a mixed state given by the density matrix ρ 1 = a k 2 P gk (2) k here P gk denotes the one dimensional projection in H 1 that projects to the one dimensional subspace spanned by element g k and T r 1 is the trace in H 1. The range rng(ρ 1 ) of the density matrix ρ 1 is spanned by those g k for which a k 0. Schrödinger finds such instantaneous change in system S 1 s state as a result of measurement on the spatially distant system S 2 already troublesome enough; yet, in [15] he goes even further by showing that if h i is another set of orthogonal unit vectors in H 2 corresponding to eigenvectors of observable H with eigenvalues λ H i, then the state Φ(x, y) can be re-written as [ ] Φ(x, y) = w i α ik g k (x) h i (y) (3) i k with constants w i and α ik depending on the set {h i } and such that the functions g i = α k ikg k (x) are normalized (but not orthogonal) and belong to rng(ρ 1 ). Schrödinger points out that by a suitable choice of h i every unit vector in rng(ρ 1) can be obtained as a g i; on the other hand, if one carries out a measurement of H on S 2 and finds eigenvalue λ H i then one has to assign state g i to system S 1. This means that by choosing an appropriate observable H to measure on system S 2 one can transform the state of system S 2 into any state of S 1 that lies in the range of ρ 1 (this transformation occurs with probability w i 2 ). Therefore, if Ψ is such that none of the a k is equal to zero, and, consequently, rng(ρ 1 ) = H 1, then... in general a sophisticated experimenter can, by a suitable device which does not involve measuring non-commuting variables, produce a non-vanishing probability of driving the system [S 1 ] into any state he chooses... [15][p. 446] 4. Indubitably the situation described here is, in present quantum mechanics, a necessary and indispensable feature. The question arises, whether it is so in Nature too. I am not satisfied about there being sufficient experimental evidence for that. Years ago I pointed out [Schrödinger s footnote: Annalen der Physik (4), 83 (1927), 961. Collected Papers (Blackie and Son, 1928), p. 141.] that when two systems separate far enough to make it possible to experiment on one of them without interfering with the other, they are bound to pass, during the process of separation, through stages which were beyond the range of quantum mechanics as it stood then. For it seems hard to imagine a complete separation, whilst the systems are still so close to each other, that, from the classical point of view, their interaction could still be described as an unretarded actio in distans. And ordinary quantum mechanics, on account of its thoroughly unrelativistic character, really only deals with 3

4 the actio in distans case. The whole system (comprising in our case both systems) has to be small enough to be able to neglect the time that light takes to travel across the system, compared with such periods of the system as are essentially involved in the changes that take place. Though in the mean time some progress seemed to have been made in the way of coping with this condition (quantum electrodynamics), there now appears to be a strong probability (as P. A. M. Dirac [Schrödinger s footnote: P.A.M. Dirac, Nature. 137 (1936), 298] has recently pointed out on a special occasion) that this progress is futile. [15][p. 451] 3 Von Neumann s reply to Schrödinger In his letter to Schrödinger (April 11, 1936) von Neumann reacts to the above passage in Schrödinger s paper: I cannot accept your 4. completely. I think that the difficulties you hint at are pseudo-problems. The action at distance in the case under consideration says only that even if there is no dynamical interaction between two systems (e.g. because they are far removed from each other), the systems can display statistical correlations. This is not at all specific for quantum mechanics, it happens classically as well. (von Neumann to Schrödinger, April 11, 1936) [20] To illustrate his point, von Neumann gives the following simple example in the letter: Let S 1 and S 2 be two boxes. One knows that 1,000,000 years ago either a white ball had been put into each or a black ball had been placed into each but one does not know which color the balls were. Subsequently one of the boxes (S 1 ) was buried on Earth, the other (S 2) on Sirius. So one has this probability distribution: Or for S 2 only: S 1 white S 2 white 1 S 2 black 0 S 2 S 2 S 1 black white black Now one digs S 1 on Earth out, opens it and sees: the ball is white. This action on Earth changes instantaneously the S 2 statistic on Sirius to S 2 white 1 S 2 black 0 4

5 (The situation reminds one of the well-known joke: Au moment même [Von Neumann s insertion: Als mit seiner Frau etwas in Paris geschah. ] mon Colonel était cocu á Madascar. (von Neumann to Schrödinger, April 11, 1936), [20] Together with von Neumann s insertion the French sentence reads something like The moment something happened to his wife in Paris the colonel was cuckolded in Madagascar. Von Neumann also reacts to Schrödinger s skeptical remark about the prospects of relativistic quantum field theory: And of course quantum electrodynamics proves that quantum mechanics and the special theory of relativity are compatible philosophically quantum electrodynamic fails only because of the concrete form of Maxwell s equations in the vicinity of a charge. (von Neumann to Schrödinger, April 11, 1936), [20] 4 Comments on von Neumann s reply There are two different ideas in von Neumann s reply to Schrödinger by which von Neumann avoids a potential clash between presence of distant correlations and the principle of no action at a distance : One is to interpret probabilities subjectively as measures of lack of knowledge. If the probabilities are just measures of our ignorance then no real, physical change takes place when the probability distribution on S 2 changes as a result of manipulation on S 2. (This idea is displayed nicely by the joke von Neumann recalls: the real physical change occurs only in Madagascar; the colonel s getting cuckolded is just a semantic change.) It is difficult however to accept a subjective interpretation of probability in physics because it is a fact that probability statements are verified in quantum mechanics by counting frequencies. The other idea is that presence of correlations between spatially separated quantum systems is not, in and by itself, reason for concern as long as one has an acceptable explanation of those correlations. In the example the explanation is the action of putting the white balls into the boxes. This action together with the subsequent observations on Earth and Syrius can be translated into probabilistic terms in the following way: Let A, A, B, B and C, C be the following events: A A B B C C white ball observed in box upon opening box on Earth black ball observed in box upon opening box on Earth white ball observed in box upon opening box on Syrius black ball observed in box upon opening box on Syrius placing white balls in both boxes on Earth placing black balls in both boxes on Earth If p(a) = p(b) = 1/2 and p(a ) = p(b ) = 1/2 are the initial probabilities of the respective events, then A and B are positively correlated: 5

6 1 2 = p(ab) > p(a)p(b) = = 1 4 Furthermore we have (4) p(ab C) = p(a C) = p(b C) = 1 (5) p(ab C ) = p(a C ) = p(b C ) = 0 (6) Consequently, the following conditions hold: p(ab C) = p(a C)p(B C) (7) p(ab C ) = p(a C )p(b C ) (8) Definition 1 Given a positive correlation p(a C) > p(a C ) (9) p(b C) > p(b C ) (10) p(ab) > p(a)p(b) (11) between events A and B in a probability space (S, p) with Boolean algebra S and probability measure p on S, an event C S is called a common cause of the correlation (11) if it satisfies conditions (7)-(10). This definition of common cause is due to H. Reichenbach [12], the event C having the properties (7)-(10) is called a (Reichenbachian) common cause. Reichenbach also seems to have embraced what came to be called the Common Cause Principle: If events A, B are correlated then the correlation is either due to a causal influence between the correlated events or there exists a common cause of the correlation. (Reichenbach himself never formulated the principle explicitly, this was articulated later, especially by W. Salmon [13]). The intuitive reason why one does not regard the correlation in von Neumann s example problematic is thus that in the example the act of placing the white balls into the boxes serves as a natural common cause of the correlation, and such a common cause explains the correlation in the sense of entailing it. Viewed from this perspective, the intended message of von Neumann s example seems to be that in the case of quantum correlations Schrödinger is considering the same should be the case; that is to say, von Neumann seems to take the position that the correlations entailed by entanglement also might have an explanation in terms of (Reichenbachian) common causes. Problem is that, unlike in von Neumann s simple example, in the case of correlations arising from entanglement, there are no obvious candidates for events that could qualify as common causes. Lack of common cause candidate events has led to the suspicion that such events cannot exist at all. Indeed, there exist a number of proofs in the literature to the effect that common cause events of correlations entailed by entanglement cannot exist: Van Fraassen was the first to link Reichenbach s notion of common cause to the EPR correlations [18] and he concluded that no common cause explanation of EPR correlations is possible. Since Fraassen s work 6

7 it has become generally accepted that the quantum correlations violate the Common Cause Principle. A careful look at the problem shows however that it is more difficult to rule out common causes of correlations than one would have thought. Let (S, p) be a classical probability space and A, B S two events that are correlated in p. The Common Cause Principle says that if A and B are causally independent, then there has to exist a common cause of the correlation. The crucial observation that makes an attempt to falsify the Common Cause Principle very difficult is that the Principle does not require the common cause to be part of the event structure S: it may very well be the case that there is no event C in S that qualifies as a common cause of the correlation between A and B but this fact in and by itself does not entail any violation of the Common Cause Principle, for it may very well be the case that S is just too small, and there might exist hidden common causes of the given correlation hidden in the sense of belonging to an event structure S which is larger than S. It turns out that such a defence of the Common Cause Principle is always possible, for one has the following result [2]: Proposition 1 Given any classical probability space (S, p) and a correlation Corr p (A, B) = p(ab) p(a)p(b) > 0 in it, there exists an extension (S, p ) of (S, p) such that there exists an event C in S which is the common cause of the correlation Corr p(a, B) > 0. (Note that (S, p ) is an extension of (S, p) if there exists a Boolean algebra homomorphism h: S S such that p (h(x)) = p(x) for every X S.) Proposition 1 entails that one can only hope to be able to falsify the Common Cause Principle if one requires of the common cause to satisfy some additional conditions that are not part of the definition of common cause. The additional conditions are typically locality conditions : probabilistic independence conditions intended to express in probabilistic terms consequences of relativistic locality (causality) principles. As can be expected, the question of whether local common cause type explanations of EPR correlations are possible, depends very sensitively on how the locality conditions are formulated: It turns out that under some formulations of those additional locality conditions hidden common causes still cannot be ruled out [17]. If one formulates more stringent locality conditions, then it is an open problem whether such strongly local hidden common causes can exist (see the review [10]). Recently, new locality conditions have been suggested that seem to exclude common cause explanations of EPR correlations (see [3]), a critical evaluation of these locality conditions is yet to be carried out, however. Once one starts talking about notions of relativity theory in connection with the discussion of quantum correlations, the proper framework in which one should deal with the problem of distant quantum correlations is relativistic quantum field theory. A specific approach to quantum filed theory is local, algebraic relativistic quantum field theory (ARQFT), axioms of which were worked out during the late fifties (see Haag s book [5] for a comprehensive presentation of the theory). It turned out that correlations between spacelike separated observables is even more endemic in local quantum field theory than in non-relativistic quantum mechanics 7

8 (see the review paper [16] and [6] for a more recent result). Presence of those spacelike correlations in relativistic quantum field theory raises the problem of the status of the Common Cause Principle in physics in an even more dramatic way because relativistic quantum field theory is a physical theory which, by its very construction, is supposed to be complying with locality and causality principles as these are understood in the spirit of the special theory of relativity. To formulate precisely the question of whether quantum field theory satisfies the Common Cause Principle and to recall the only result known in this connection, we need some definitions: Let N be a von Neumann algebra, P(N ) its projection lattice and φ a normal state on N. Recall that two elements A, B P(N ) are called compatible if there exists a distributive sublattice of P(N ) containing both A and B. Definition 2 Let A, B P(N ) be two compatible elements that are correlated in φ: φ(a B) > φ(a)φ(b) (12) C P(N ) is a common cause of the correlation (12) if C is compatible with both A and B and the following conditions (completely analogous to (7)-(10)) hold φ(a B C) φ(c) φ(a B C ) φ(c ) φ(a C) φ(c) φ(b C) φ(c) = φ(a C) φ(b C) φ(c) φ(c) = φ(a C ) φ(b C ) φ(c ) φ(c ) > φ(a C ) φ(c ) > φ(b C ) φ(c ) (13) (14) (15) (16) For a point x in the Minkowski space M let BLC(x) denote the backward light cone of x; furthermore for an arbitrary spacetime region V let BLC(V ) x V BLC(x). For spacelike separated spacetime regions V 1 and V 2 let us define the following regions wpast(v 1, V 2) (BLC(V 1) \ V 1) (BLC(V 2) \ V 2) (17) cpast(v 1, V 2 ) (BLC(V 1 ) \ V 1 ) (BLC(V 2 ) \ V 2 ) (18) spast(v 1, V 2) x V1 V 2 BLC(x) (19) Obviously it holds that spast(v 1, V 2 ) cpast(v 1, V 2 ) wpast(v 1, V 2 ) (20) Definition 3 Let {N (V )} be a net of local von Neumann algebras over Minkowski space satisfying the standard axioms of ARQFT (isotony, Einstein locality, Poincaré covariance, weak additivity, spectrum condition). Let V 1 and V 2 be two spacelike separated spacetime regions, and let φ be a locally normal state on the quasilocal algebra A. If for any pair of projections A N (V 1 ) and B A(V 2 ) it holds that if φ(a B) > φ(a)φ(b) (21) 8

9 then there exists a projection C in the von Neumann algebra N (V ) which is a common cause of the correlation (21) in the sense of Definition 2, then the local system is said to satisfy Weak Common Cause Principle: if V wpast(v 1, V 2 ) Common Cause Principle: if V cpast(v 1, V 2 ) Strong Common Cause Principle: if V spast(v 1, V 2) We say that Reichenbach s Common Cause Principle holds for the net (respectively holds in the weak or strong sense) iff for every pair of spacelike separated spacetime regions V 1, V 2 and every normal state φ, the Common Cause Principle holds for the local system (N (V 1), N (V 2), φ) (respectively in the weak or strong sense). Problem: Does any of the above Common Cause Principles hold in quantum field theory? If V 1 and V 2 are complementary wedges then spast(v 1, V 2 ) =. Since the local von Neumann algebras pertaining to complementary wedges are known to contain correlated projections (see [16]), the Strong Reichenbach s Common Cause Principle trivially fails in AQFT. The question of whether Reichenbach s Common Cause Principle holds in AQFT was first formulated in [8] (see also [9]) and the answer to it is not known. What is known is that the Weak Reichenbach s Common Cause Principle typically holds under mild assumptions on the local net {N (V )}: Proposition 2 If a net {N (V )} with the standard conditions (isotony, Einstein locality, Poincaré covariance, weak additivity, spectrum condition) is such that it also satisfies the local primitive causality condition and the algebras pertaining to double cones are type III, then every local system (N (V 1), N (V 2), φ) with V 1, V 2 contained in a pair of spacelike separated double cones and with a locally normal and locally faithful state φ satisfies Weak Reichenbach s Common Cause Principle. (See [11] for the proof of the above proposition and for additional analysis of the status of Reichenbach s Common Cause Principle in quantum field theory.) Local primitive causality is a condition that expresses the hyperbolic character of time evolution in AQFT. For a spacetime region V let V = (V ) denote the causal completion (also called causal closure and causal hull) of V, where V is the set of points that are spacelike form every point in V. The net {N (V )} is said to satisfy the local primitive causality condition if N (V ) = A(V ) for every nonempty convex region V. 5 Concluding remarks Von Neumann s letter to Schrödinger is probably the first formulation of Reichenbcah s Common Cause Principle in connection with quantum correlations but an implicit formulation only since the technically explicit notion of common cause does not appear in the letter. It also is worth 9

10 pointing out that, apparently, von Neumann did not see a major difference between quantum and classical correlations from the perspective of the Common Cause Principle. Once however the notion of common cause is specified in the sense of Reichenbach s definition, the concept of common cause and the status of the associated Common Cause Principle can be subjected to a rigorous analysis. The analysis has led to a number of precise problems, some of which are still open. Specifically, it is not known whether relativistic quantum field theory is causally rich enough to be able to give a local common cause explanation of the spacelike correlations it predicts. Nor has it been proven in full generality that local common causes of standard EPR correlations cannot exist. The difficulty with proving precise theorems concerning the Common Cause Principle is that the notion of (Reichenbachian) common cause is rather subtle, hence problems relevant for the status of the Principle are non-trivial. It is difficult even to decide whether a given probability space (S, p) is causally closed in the sense that it contains a common cause of every correlation between correlated events A, B that are causally independent, R ind (A, B). It turns out that causal closedness of (S, p) with respect to a causal independence relation R ind depends sensitively on both (S, p) and R ind, and, while causal closedness is not impossible it is possible even if the event structure S contains a finite number of elements [4] it does not always hold. There does not seem to exist a canonical procedure by which causal closedness could be verified, and there are a number of open questions concerning causal closedness (see [4]). Acknowledgement: I wish to thank Marina von Neumann Whitman for her permission to quote from von Neumann s letter to Schrödinger. Work supported by OTKA (contract number: T 43642) and by Alexander von Humboldt Foundation (through a Sofja Kovalevskaja Award). References [1] A. Einstein, B. Podolsky, N.Rosen: Can Quantum Mechanical Description of Physical Reality Be considered Complete? Physical Review 47 (1935) [2] G. Hofer-Szabó, M. Rédei and L.E. Szabó: On Reichenbach s Common Cause Principle and Reichenbach s notion of common cause The British Journal for the Philosophy of Science 50 (1999) [3] G. Grasshoff, S. Portmann and A Wüthrich: Minimal assumption derivation of a Bell-type inequality The British Journal for the Philosophy of Science (forthcoming) [4] B. Gyenis and M. Rédei: When can statistical theories be causally closed? Foundations of Physics 34 (2004) [5] R. Haag: Local Quantum Physics (Springer Verlag, Berlin, 1992) 10

11 [6] H. Halvorson and R. Clifton: Generic Bell correlation between arbitrary local algebras in quantum field theory Journal of Mathematical Physics 41 (2000) [7] M. Jammer: The EPR Problem in its Historical Development in P. Lahti and P.Mittelstaedt (eds), Symposium on the Foundations of Modern Physics. 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment (Singapore: World Scientific, 1985) [8] M. Rédei: Reichenbach s common cause principle and quantum field theory Foundations of Physics 27 (1997) [9] M. Rédei: Quantum Logic in Algebraic Approach (Kluwer Academic Publishers, Dordrecht, 1998) [10] M. Rédei: Reichenbach s Common Cause Principle and quantum correlations in Modality, Probability and Bell s Theorems, NATO Science Series, II. Vol. 64., T. Placek and J. Butterfield (eds.) (Kluwer Academic Publishers, Dordrecht, Boston, London, 2002) [11] M. Rédei and S.J. Summers: Local primitive causality and the Common Cause Principle in quantum field theory Foundations of Physics 32 (2002) [12] H. Reichenbach: The Direction of Time (University of California Press, Los Angeles, 1956) [13] W.C. Salmon: Scientific Explanation and the Causal Structure of the World (Princeton University Press, Princeton, 1984) [14] E. Schrödinger: Discussion of probability relations between separated systems Proceedings of the Cambridge Philosophical Society 31 (1935) [15] E. Schrödinger: Probability relations between separated systems Proceedings of the Cambridge Philosophical Society 32 (1936) [16] S.J. Summers: On the independence of local algebras in quantum field theory, Reviews in Mathematical Physics 2 (1990) [17] Szabó, L.E.: Attempt to resolve the EPR-Bell paradox via Reichenbach s concept of common cause International Journal of Theoretical Physics 39 (2000) [18] B.C. Van Fraassen: The Charybdis of Realism: Epistemological Implications of Bell s Inequality in J. Cushing and E. McMullin (eds.), Philosophical Consequences of Quantum Theory (Notre Dame: University of Notre Dame Press, 1989) [19] J. von Neumann: Mathematische Grundlagen der Quantenmechanik (Dover Publications, New York, 1943) (first American Edition; first edition: Springer Verlag, Heidelberg, 1932; first English translation: Princeton University Press, Princeton, 1955.) [20] J. von Neumann to E. Schrödinger (April 11, 1936), translated by M. Rédei, forthcoming in [21] [21] J. von Neumann: Selected Letters, M. Rédei ed., (American Mathematical Society, Providence, Rhode Island) (forthcoming) 11