Isomorphism between the Peres and Penrose Proofs of the BKS Theorem in Three Dimensions

Abstract

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.Keywords Kochen-Specker theorem · Bell's theorem · Foundations of quantum mechanics Some time back Peres [1] gave a proof of the Bell-Kochen-Specker (BKS) theorem [2, 3] using 33 real rays (or directions) in three dimensions. Penrose [4] later gave a different proof of the theorem using 33 complex rays in three dimensions. Penrose pointed out that his set of rays is essentially complex (i.e., there is no basis in which the components of all the rays can be made real) and that there is no Hilbert space rotation that will take his rays into those of Peres. It might therefore be thought that the proofs of the BKS theorem based on the two sets of rays are essentially different. However we show in this paper that the Kochen-Specker diagrams of the Peres and Penrose rays are identical, and that they therefore furnish isomorphic proofs of the BKS theorem. We exploit the common cubic symmetry of both sets of rays to give a unified proof of the BKS theorem for them that is shorter than the one given by Peres. Finally, we demonstrate that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.The 33 rays of Peres can be visualized in terms of the geometry of a cube. Each ray goes out from the center of a cube to a point on its surface as follows: three

How orthogonalities set Kochen-Specker sets

“…There are no obvious trends in the data shown in table 1 to account for this surprising lack of parameters. This finding adds a curious apparent uniqueness to the Peres and Penrose sets in H 3 and thus strengthens the appeal of the isomorphism found in [10].…”

“…It is easy to see there can be no unitary operation to reconcile the sets as the Penrose set has vector entries from C 3 and cannot be confined to a real subspace while the Peres set is entirely real. Furthermore, both sets are shown in [10] to be special cases of a more general 3-parameter family of 33 vectors.…”

“…Most notably, in a 3-dimensional Hilbert space, the original KS proof of 117 vectors has been replicated with sets using between 31 and 33 vectors [2,3,4,5] and in a 4-dimensional Hilbert space, sets of between 24 and the lower limit of 18 vectors have been realised [2,6,7]. Additionally, an extensive computer search has recently revealed over one thousand of such KS sets in R 4 [8].In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have…”

“…In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have identical orthogonality conditions but are not unitarily equivalent.…”

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

“…(Gould and Aravind have proven that the so-called Penrose's 3-dim KS set is isomorphic to Peres' one [69].) It is well-known that all of them are critical and our program states01 confirms that.…”