Super-Quantum, Non-Signaling Correlations Cannot Exist Pierre Uzan University Paris-Diderot laboratory SPHERE, History and Philosophy of Science Abstract It seems that non-local correlations stronger than quantum correlations between two non-signaling systems cannot exist for the simple reason that the property of non-signaling implies that the CHSH correlation factor must satisfy the Tsirelson bound. This result is obtained after translating the fashionable language of boxes into the language of observables, which is used to compute the CHSH correlation factor. A recently proposed toy-model of such nonsignaling, super-quantum correlations is then considered and it is shown that it suffers from logical inconsistency. Keywords: Super-quantum correlations; Tsirelson bound; PR-boxes. Introduction Some physicists have recently suggested that non-local correlations stronger than quantum correlations between two sub-systems that cannot exchange any signal would be theoretically possible [1]. As noted by these same authors, such superquantum correlations have not yet been observed. Moreover, some indirect arguments have been formulated against the existence of these correlations, by appealing to a causality principle [2] or by mentioning its rather implausible consequences [3]. However, it seems that the absence of experimental data and the latter counter-arguments is not enough to fully convince these physicists of the inexistence of non-signaling, super-quantum correlations and the latter continue to explore very seriously their extraordinary consequences, like for example the possibility of non-local computation [1]. The existence of these correlations, which then remains very speculative for 1
now, has been suggested from the possibility of realizing specific, ad-hoc correlation functions between observables, correlations functions that have been more recently translated into the language of boxes. A box can be described by an arithmetic relation between couples of inputs and couples of outputs. In particular, PR-boxes (labeled from the initials of Popescu and Rohrlich), which are defined by the following relation between its binary, 0 or 1, inputs (x,y) and its binary, 0 or 1, possible outputs (a,b): a + b = x.y, where the addition + is modulo 2 would give rise to such super-quantum correlations for which the CHSH correlation factor violates the Tsirelson bound [4]. In addition, as explained by Popescu [1], the non-signaling property can be implemented by asserting that these PR-boxes yield couples of outputs with equal probabilities for each of the possible values of the product x.y (since, in this case, one of the parties cannot exploit his/her local information to find out the input of the other party). Explicitly, for x.y = 1, both inputs being equal to 1, the two possible outputs (a = 0 and b =1) or (a =1 and b = 0) must have a probability of occurrence of ½; and when x.y = 0, which means that the two inputs are different, the two possible outcomes (a = 0 and b = 0) or (a =1 and b=1) must also have a probability of occurrence of ½. Non-signaling PR-boxes are then compactly described by the following relation [5]: (C0) P (a, b / x, y) = ½ if a + b (mod 2) = x.y is realized = 0 otherwise. Toy-models that are supposed to satisfy such condition, and which would then show that bipartite super-quantum correlations can really be observed between two non-signaling systems, have even been proposed in recent publications [6-7]. The present article will directly questions the logical consistency of the very idea of super-quantum, non-signaling correlations. It will first show from very simple arguments formulated in the standard, quantum 2
language of observables into which the fashionable language of boxes can always be translated, that super-quantum correlations cannot exist between non-signaling systems. Then, in order to clear up a persistent ambiguity, the reference toy-model of such non-signaling PR-boxes proposed by Popescu and Rohrlich will be briefly analyzed. It will be shown that this model cannot do the job because it is based on an inconsistent correlation function. I. Superquantum, non-signaling correlations cannot exist. To show that non-local correlations stronger than quantum correlations between two non-signaling systems cannot exist by principle, let us focus separately on the two essential characteristics of such superquantum correlations: (C1) The assertion that these correlations are more non-local than quantum correlations is expressed by the condition that the CHSH correlation factor for two couples of observables (A 0, A 1 ) and (B 0, B 1 ) respectively defined on two sub-systems S1 and S2 (and for any state of the composed system S1+S2): R R {(A 0, A 1 ); (B 0, B 1 )} = < A 0 B 0 + A 0 B 1 + A 1 B 0 - A 1 B 1 > is strictly greater (in absolute value) than the Tsirelson bound: (1) R > 2 2 (C2) The assertion that S1 and S2 cannot exchange signals (regarding these two couples of observables) is, in the language of boxes, expressed by the conditions that the probability that Alice obtains a particular outcome a is independent from Bob s input and vice versa [1]: b P(a, b / x, y) = P(a / x), a P(a, b / x, y) = P(b / y). 3
Note that the first of these equalities can be more simply re-written by noting the sum on the possible outcomes b corresponding to the input y (measurement of B 0 or B 1 ) in the left member by P (a / x, y): (2) P (a / x, y) = P (a / x) Similarly, in the symmetric condition (3) the sum on the possible outputs a corresponding to the input x (measurement of A 0 or A 1 ) defines P (b / x, y): (3) P (b / x, y) = P (b / y) The condition (C1) is expressed in the language of observables while the condition (C2) refers to probabilistic relations between inputs and outputs of Alice s and Bob s boxes. In order to show our main result, we have first to make explicit the correspondence between these two languages. In spite of the underlying hypothesis [1] that such super-quantum, non-signaling correlations would rely on hypothetical theories that supposedly could not be expressed within the formalism of quantum theory, nothing prevent us a priori to use this formalism if it can successfully describe the functioning of PR-boxes. No super-principle, which would remain to justify, can forbid the use of a mathematical formalism if the latter gives a correct account of a situation. To say in the language of boxes that Alice chooses the input x (for x = 0 or 1) and obtains the output a (for a = 0 or 1) can be translated in the language of observables by saying that Alice is measuring the observable A x (with x = 0 or 1) whose eigenvalues are 0 and 1, where the observables A x are supposed to act on a two-dimensional Hilbert space H A spanned by their two eigenvectors > Ax and > Ax. Similarly, to say in the language of boxes that Bob chooses the input y (with y = 0 or 1) and obtains the output b (with b = 0 or 1) can be translated into Bob measures the observable B y (with y = 0 or 1) whose eigenvalues are 0 and 1, where B y is supposed to act on a two-dimensional Hilbert space H B spanned by its two eigenvectors > By and > By. 4
Now, the CHSH correlation factor R relative to these couples of boxobservables (A 0, A 1 ) and (B 0, B 1 ) is defined as follows: R = < A 0 B 0 + A 0 B 1 + A 1 B 0 - A 1 B 1 >. Note that this definition of R, which is in conformity with condition (C1), is meaningful for the game of boxes provided that the observables A x (for x = 0 or 1) and B y (for y = 0 or 1) are defined as explained above. The incompatibility of the two conditions (C1) and (C2) that characterize non-signaling super-quantum correlations will now be established. Let us first show that the condition (C2) of non-signaling implies the commutation of all the observables defined for Alice s box (which are here A 0 and A 1 ) with all the observables defined for Bob s box (which are here B 0 and B 1 ), that is: [A 0, B 0 ] = [A 0, B 1 ] = [A 1, B 0 ] = [A 1, B 1 ] = 0. The condition (2) means that Alice s probability for obtaining the outcome a by measuring the observable A x (with x = 0 or 1) is independent of the fact that Bob measures B y (with y = 0 or 1) with the outcome b. It can be rewritten as: (2 ) P (A x = a / B y = b) = P (A x = a). Similarly, the condition (3) can be translated in the language of observables as: (3 ) P (B y = b / A x = a) = P (B y = b). From the conditions (2 ) and (3 ), we can deduce that: P (A x = a / B y = b). P (B y = b) = P (B y = b / A x = a). P (A x = a). The left member of this equation evaluates Alice s probability of obtaining the outcome a for A x once Bob has obtained the outcome b by measuring B y. It is then the sequential probability to measure B y = b and then A x = a. 5
Similarly, the right member of this equation evaluates the sequential probability to measure A x = a and then B y = b. Consequently, the non-signaling conditions (2) and (3) imply that: P (B y = b and then A x = a) = P (A x = a and then B y = b). Since this condition holds for all possible outputs a and b, which are the possible outcomes of the measurement of A x and B y, and for all possible inputs x and y, it can be concluded that the non-signaling condition (C2) implies that all Alice s observables commute with all Bob s observables: (4) [A 0, B 0 ] = [A 0, B 1 ] = [A 1, B 0 ] = [A 1, B 1 ] = 0. The second step of the argument is to show that the condition (4) of commutativity of all Alice s observables with all Bob s observables is sufficient for establishing the corresponding CHSH factor R defined above is smaller than the Tsirelson bound. This derivation is nothing but an adaptation to the present case of a well-known theorem shown by Landau [8] for binary observables with possible values -1 or +1. This theorem shows that the condition (4) entails that the CHSH correlation factor R is less than the Tsirelson bound. Actually, it is easy to see that this conclusion also holds with the weaker assumption that the spectrum of all the observables is bounded by 1 (that is, the possible values a and b of A x and B y are such that a 1 and b 1, for all x and y) 1 and, in particular, in the present case where the values of the observables can be 0 or 1: R 2 2. Consequently, the condition (C1) of super non-locality (violation of the T- bound), which is expressed by the inequality (1), is incompatible with the non- 1 For, in this case, the quantities c and d considered by Landau in his derivation are such that (with his notations): c 2 4 + [a1, b1] [b2, a2] and d 2 4 + [a1, b1] [a2, b2], the equalities being replaced by inequalities. Consequently, the result < c > 2 < c 2 > 8 still holds. 6
signaling condition (C2), which implies the inequality (5). Correlations that violate the T-bound can exist only in the case the two sub-systems are signaling. 2. What about the explicit models of non-signaling PR-boxes? In contradiction with this result, some explicit models of boxes that would satisfy the conditions (C1) and (C2) at once have been proposed [6-7]. The solution of this paradoxical situation is that these toy-models are either impossible to realize for logical reasons or they actually allow the exchange of signals. Let us briefly consider one of them, which is taken in the current literature as the reference model of such super-quantum, non-signaling correlations. This model has been proposed by Rohlich and Popescu [6]: Rohlich and Popescu have built an ad-hoc correlation function between two couples of spin observables (A, A ) and (B, B ), respectively defined on two non-signaling sub-systems S1 and S2, and whose directions a, b, a, b are separated by successive angles of /4 in a same plane (see the diagram, where the direction of A has been conventionally fixed to 0): B = S 3 /4 A = S /2 B = S /4 A = S 0 Representation in the same plane of the two couples of spin observables (A,A ) and (B,B ) In this model the correlation function between two spin observables respectively defined on S1 and S2 only depends on the angle between these two observables. In particular, the expectation values of the product of two spin observables, which depend only of their angle, is defined as follows: 7
E( /4) = +1 and E(3 /4) = -1, Then, due to the directions of the spin observables reported in the previous diagram, the CHSH correlation factor can be computed as: (6) R = 3 E( /4) - E(3 /4) = 4. This result would then show that non-local correlations with R > 2 2 can exist between no-signaling systems. However, if we consider a separable state of the composite system S, the expectation values of the products of observables involved in the different terms of the relevant CHSH correlation factor are, in this case, factorable: they can then be written as the product of the expectation values of the observables acting on S1 an S2, respectively. Taking now into account the values of the correlation function for the different terms of R, we obtain the following equations: <AB> = <S /2 I > < I S /4 > = +1 <AB > = <S /2 I > < I S /4 > = +1 <A B> = <S I > < I S /4 > = +1 <A B > = <S I > < I S /4 > = -1. It is easy to see that the fourth equation, which implies that <S I > and <I S /4 > are of opposite signs, is incompatible with the first three ones, which imply the contrary. Consequently, this model is logically inconsistent. Conclusion The dream of realizing non-signaling PR-boxes, which would give rise to extraordinary, if not magical consequences, has resisted to credible counterarguments (relative to causality and to the implausibility of its consequences). 8
But this article has shown that the very idea of existence of such non-signaling super-quantum correlations is not logically consistent. This result has been derived from very simple, logical arguments and by clarifying the rather hermetic language of boxes. We hope that this stronger result will definitely convince the community of physicists of the impossibility of non-signaling super-quantum correlations and then of the unreality of non-signaling PRboxes. If it is not yet the case, the belief in the existence of such correlations can be compared with the fierceness of building a perpetual motion machine against the laws of thermodynamics. References 1. S. Popescu 2015, Non-locality beyond quantum mechanics, in Alisa Bokulich, Gregg Jaeger Editors, Philosophy of Quantum Information and Entanglement, pp. 3-15, Cambridge Books Online. 2. M.P. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter and M. Zukowski 2010, Information Causality as a Physical Principle. arxiv:0905.2292v3, 6 May. 3. W. van Dam 2005, Implausible Consequences of Superstrong Nonlocality, arxiv:quant-ph/0501159. 4. J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt 1969, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 884. 5. J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu and D. Roberts 2005, Non-local correlations as an information theoretic resource, Physical Review A 71. 6. S. Popescu and D. Rohrlich 1994, Quantum nonlocality as an axiom. Found. Phys. 24: 379-385. 9
7. T. Filk, A 2015, Mechanical model of a PR-Box, arxiv:1507.06789. 8. L.J. Landau 2001, Quantum Theory: Mathematics and Reality. Chap 7-8. Talk given at the King's College History of Mathematics Summer School, June 27-28 2001. Mathematics Department, King's College London. 10