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On coloring the rational quantum sphere Hans Havlicek Institut für Geometrie, Technische Universität Wien Wiedner Hauptstraße 8-10/1133, A-1040 Vienna, Austria [email protected] and Günther Krenn and Johann Summhammer Atominstitut der österreichischen Universitäten, Stadionallee 2 A-1020 Vienna, Austria [email protected], [email protected] and Karl Svozil Institut für Theoretische Physik, Technische Universität Wien Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria e-mail: [email protected] (to whom correspondence should be directed) http://tph.tuwien.ac.at/e svozil/publ/colo.{htm,ps,tex} 1 Abstract We discuss types of colorings of the rational quantum sphere similar to the one suggested recently by Meyer [1], in particular the consequences for the Kochen-Specker theorem [2] and for the correlation functions of entangled subsystems. Recently, Godsil and Zaks [3] published a constructive coloring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere, exactly one ray is red, white and black. They also showed that any consistent coloring of the real sphere requires an additional color. Based on this result, Meyer [1] suggested that the physical impact of the Kochen-Specker theorem [2] is “nullified,” since for all practical purposes it is impossible to operationalize the difference between any dense set of rays and the continuum of Hilbert space rays. We shall argue here that Meyer’s result is itself “nullified” for a variety of reasons: (i) The Kochen-Specker theorem deals with the nonembedability of certain partial algebras—in particular the ones arising in the context of quantum mechanics—into total Boolean algebras. 0−1 colorings serve as an important method of realizing such embeddings: e.g., if there are a sufficient number of them to separate any two elements of the partial algebra, then embedability follows [2, 4, 5]. Kochen and Specker’s original paper [2] contains a much stronger result—the nonexistence of 0 − 1 colorings—than would be needed for nonembedability. (The above considerations are explicitly mentioned by them.) Yet the mere existence of some homomorphisms is a necessary but no sufficient criterion for embedability. This has already be pointed out by Clifton and Kent [6]. To put it pointedly: Meyer is confusing necessity with sufficiency. 2 (ii) Within the Birkhoff-von Neumann scheme of quantum logic [7], the set of propositions is not closed under quantum logical operations; in particular under the nor-operation. (This argument is not valid in the Partial Algebra approach developed by Kochen and Specker [8, 9, 10, 2].) (iii) The regress to unsharp measurements is rather questionable and would allow total arbitrariness in the choice of approximation. (iv) The probabilities resulting from these truth assignments are noncontinuous and arbitrary. Squares of even and odd numbers are either 0 and 1 modulo 4, respectively. Thus, for any point of the rational unit sphere S 2 ∩Q3 = x1 x2 x3 , , n n n | x21 + x22 + x23 = n2 , x1 , x2 , x3 ∈ Z, n ∈ N and odd just two of the xi in x21 +x22 +x23 = n2 are even and one is odd. From any three orthogonal points, exactly one point has an odd first (or, alternatively, second and third) component. This is due to the fact that, if two points (x1 , x2 , x3 ) and (y1 , y2 , y3 ) are orthogonal, then x1 y1 + x2 y2 + x3 y3 = 0, and these points must differ in which component is odd. Therefore, a 0−1 coloring is obtained by associating with (x1 /n, x2 /n, x3 /n) the parity of x1 . (Recall that two of the components x1 , x2 , x3 are even and one is odd.) Such a coloring can also be obtained by identifying two of the three colors of Godsil and Zaks [3]. However, as has been stated before, the mere existence of a homomorphism, i.e., a 0−1 coloring, does not imply the possibility to embed the partial algebra into a total one, let aside any physically relevant hidden parameter model. This is a subtle point often disregarded: of course, if there does not even exist a homomorphism, this implies that the algebraic structure is not embedable. But the existence of a homomorphism is no sufficient criterion for embedability [2, 4, 11]. The Godzil and Zaks’ construction merely provides three homomorphisms. Embedability requires that there exist enough 3 , homomorphisms to guarantee, for example, separability. This, however, has never been shown in Meyer’s paper. Another difficulty of Meyer’s approach lies in the total arbitrariness of which approximation should be taken. That is, depending on which approximation is chosen, one predicts different results. This is even more problematic if one realizes that the each color of the 0 − 1 coloring of the rational unit sphere is dense. The rational unit sphere is not closed under certain geometrical operations such as taking two orthogonal ray of the subspace spanned by two noncollinear rays (the cross product of the associated vectors). This can be easily demonstrated by considering the two vectors 3 4 , ,0 , 5 5 4 3 0, , ∈ S 2 ∩ Q3 . 5 5 The cross product thereof is 12 −9 12 , , 6∈ S 2 ∩ Q3 . 25 25 25 Indeed, if instead of S 2 ∩ Q3 we would start with three non-orthogonal, non-collinear rational rays and generate new ones by the cross product, we would end up with all rational rays [12]. In the Birkhoff-von Neumann approach to quantum logics [7], this nonclosedness under elementary operations such as the nor-operation might be considered a serious deficiency which rules out the above model as an alternative candidate for Hilbert space quantum mechanics. This argument does not apply to the Kochen-Specker partial algebra approach to quantum logic, since there operations among propositions are only allowed if the propositions are comeasurable. Indeed, it may just be that the relative (with respect to other sets such as the rational rays) “thinness” which guarantees colorability—by not allowing the formation of finite cycles such as the ones introduced by Kochen and 4 Specker [2]—but which on the other hand does not allow closedness under the quantum logical operations. To put it differently: given any nonzero measurement uncertainty ε and any non-colorable Kochen-Specker graph Γ(0) [2, 11], there exists another graph (in fact, a denumerable infinity thereof) Γ(δ) which lies inside the range of measurement uncertainty δ ≤ ε [and thus cannot be discriminated from the non-colorable Γ(0)] which can be colored. Such a graph, however, might not be connected in the sense that the associated subspaces can be cyclically rotated into itself by local transformation along single axes. The set Γ(δ) might thus correspond to a collection of tripods such that none of the axes coincides with any other, although all of those non-identical single axes are located within δ apart from each other. Indeed, this seems to be precisely how the Clifton-Kent construction works [6]. If this is indeed the case, then the reason why a Kochen-Specker type contradiction does not occur is the impossibility to “close” the argument; to complete the cycle: the necessary elements of physical reality are simply not available in the rational sphere model. (For the same reason, an equilateral triangle does not exist in Q2 [13].) Let us re-state the physical interpretation of the coloring schemes discussed above. Any linear subspace Sp r of a vector r can be identified with the associated projection operator Er and with the quantum mechanical proposition “the physical system is in a pure state Er ” [14]. The coloring of the associated point on the unit sphere (if it exists) is equivalent with a valuation or two-valued probability measure P r : Er 7→ {0, 1} where 0 ∼ #2 and 1 ∼ #1. That is, the two colors #1, #2 can be identified with the classical truth values and with the EPR- type “elements of physical 5 reality” [15]: “It is true that the physical system is in a pure state Er ” and “It is false that the physical system is in a pure state Er ,” respectively. Since, as has been argued before, the rational unit sphere has chromatic number three, two colors suffice for a reduced coloring generated under the assumption that the colors of two rays in any orthogonal tripod are identical. This effectively generates consistent valuations or “elements of physical reality” associated with the dense subset of physical properties corresponding to the rational unit sphere [1]. For an extension of these arguments, see Kent [16]. Due to the density of points colored with any single color, any such coloring is non-continuous in the sense that between any two points of equal color there is a point (indeed, an infinity thereof) of different color. Let us then assume that any such value-definiteness might apply also to non-local setups and consider Bell-type correlation functions for spin- 12 state measurements. For singlet states along two directions which are an angle θ apart, the quantum probabilities to find identical two particle states ++ or −− is P = = P ++ + P −− = sin2 (θ/2), whereas for the non-identical states +− and −+ it is P 6= = P +− + P −+ = cos2 (θ/2). The corresponding classical quantities are P = = θ/π and P 6= = 1 − θ/π [17, 18, 19]. Since the 0 − 1 coloring is dense, then, depending on the approximation chosen, the probabilities P ++ , P +− , P −+ , P −− may take on any value between 0 and 1. This would clearly contradict the physical predictions stated above. Thus if there exists any coloring of the associated fourdimensional model, any such coloring should be in accordance with the physical findings which support quantum mechanics. In particular, any probability theory derived from the valuations of dense subsets of Hilbert spaces should be able to reproduce the well-known quantum correlations which violate value-definiteness as well as locality. This, however, is neither possible with the usual classical 6 value-definiteness, nor with the discontinuity originating in the denseness of any single color. Acknowledgments The authors would like to acknowledge stimulating discussions with Ernst Specker. We also would like to thank the Referees for suggesting many improvements. References [1] David A. Meyer. Finite precision measurement nullifies the KochenSpecker theorem. Physical Review Letters, 83(19):3751–3754, 1999. arXiv:quant-ph/9905080. [2] Simon Kochen and Ernst P. Specker. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17(1):59– 87, 1967. Reprinted in [20, pp. 235–263]. [3] C. D. Godsil and J. Zaks. Coloring the sphere. University of Waterloo research report CORR 88-12, 1988. [4] Pavel Pták and Sylvia Pulmannová. Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht, 1991. [5] Karl Svozil and Josef Tkadlec. Greechie diagrams, nonexistence of measures in quantum logics and Kochen–Specker type constructions. Journal of Mathematical Physics, 37(11):5380–5401, November 1996. [6] Rob Clifton and Adrian Kent. Simulating quantum mechanics by noncontextual hidden variables. arXiv:quant-ph/9908031 v4, 1999. 7 [7] Garrett Birkhoff and John von Neumann. The logic of quantum mechanics. Annals of Mathematics, 37(4):823–843, 1936. [8] Ernst Specker. Die Logik nicht gleichzeitig entscheidbarer Aussagen. Dialectica, 14:175–182, 1960. Reprinted in [20, pp. 175–182]; English translation: The logic of propositions which are not simultaneously decidable, reprinted in [21, pp. 135-140]. [9] Simon Kochen and Ernst P. Specker. Logical structures arising in quantum theory. In Symposium on the Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, pages 177–189, Amsterdam, 1965. North Holland. Reprinted in [20, pp. 209–221]. [10] Simon Kochen and Ernst P. Specker. The calculus of partial propositional functions. In Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, Jerusalem, pages 45–57, Amsterdam, 1965. North Holland. Reprinted in [20, pp. 222–234]. [11] K. Svozil. Quantum Logic. Springer, Singapore, 1998. [12] Hans Havlicek and Karl Svozil. Density conditions for quantum propositions. Journal of Mathematical Physics, 37(11):5337–5341, November 1996. [13] Ernst Specker. private communication, 1999. [14] John von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer, Berlin, 1932. English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. [15] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantummechanical description of physical reality be considered complete? Physical Review, 47:777–780, 1935. Reprinted in [22, pages. 138-141]. 8 [16] Adrian Kent. Non-contextual hidden variables and physical measurements. Physical Review Letters, 83(19):3755–3757, 1999. arXiv:quant-ph/9906006. [17] J. F. Clauser and A. Shimony. Bellos theorem: experimental tests and implications. Rep. Prog. Phys., 41:1881–1926, 1978. [18] Asher Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Dordrecht, 1993. [19] G. Krenn and K.Svozil. Stronger-than-quantum correlations. Foundations of Physics, 28(6):971–984, 1998. [20] Ernst Specker. Selecta. Birkhäuser Verlag, Basel, 1990. [21] Clifford Alan Hooker. The logico-algebraic approach to quantum mechanics. D. Reidel Pub. Co., Dordrecht; Boston, [1975]-1979. [22] John Archibald Wheeler and Wojciech Hubert Zurek. Quantum Theory and Measurement. Princeton University Press, Princeton, 1983. 9