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arXiv:1310.0190 · Chinese Physics Letters

State-Independent Proof of Kochen—Specker Theorem with Thirty Rank-Two Projectors

S. P. Toh

Published Oct 1, 2013• 4 citations
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Abstract

The Kochen—Specker theorem states that noncontextual hidden variable theories are incompatible with quantum mechanics. We provide a state-independent proof of the Kochen—Specker theorem using the smallest number of projectors, i.e., thirty rank-2 projectors, associated with the Mermin pentagram for a three-qubit system.
DOI
10.1088/0256-307x/30/10/100303

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Kochen—Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors was proposed in 1995 [Kernaghan M and Peres A Phys. Lett. A 198 (1995) 1 ] for a three-qubit system, there have been plenty of the same type KS sets and we propose a systematic way to produce them. We also propose a probabilistic state-dependent proof of the KS theorem that mainly focuses on the values assignment for all the rank-2 projectors.
Contextuality is one of the fundamental deviations of quantum mechanics from classical physics.The Kochen-Specker theorem shows that non-contextual classical physics with hidden variables is inconsistent with the predictions of quantum mechanics.Parity proof is one of the many different approaches applied to prove Kochen-Specker theorem for quantum system with different dimensionality.This method of proof requires Kochen-Specker sets that were composed of an odd number of bases and an even number of projectors.In most of the cases, the number of Kochen-Specker sets produced is large and its production is generally aided by computer calculation.However, manual generation of the Kochen-Specker sets are also reported previously for qutrit and two-qubit systems.We put forward the first and surprisingly simple method to generate manually Kochen-Specker sets, with respect to the Mermin pentagram, that can be used for the parity proof of the Kochen-Specker theorem in a state space of eight-dimensional three-qubit system.
A recent proposal to experimentally test quantum mechanics against noncontextual hidden-variable theories [Phys. Rev. Lett.80, 1797 (1998)] is shown to be related with the smallest proof of the Kochen–Specker theorem currently known [Phys. Lett.A212, 183 (1996)]. This proof contains eighteen yes-no questions about a four-dimensional physical system, combined in nine mutually incompatible tests. When these tests are considered as tests about a two-part two-state system, then quantum mechanics and noncontextual hidden variables make the same predictions for eight of them, but make different predictions for the ninth. Therefore, this ninth test would allow us to discriminate between quantum mechanics and noncontextual hidden-variable theories in a (gedanken) single run experiment.
The Kochen-Specker theorem (KS) states that Ilonconrextual hidden variable theories are incompatible with quantum mechanics. Since the first proof of the theorem, von Neumann projection measurement has been used and the quantum system considered has dimensionality of at least three. However, recently generalized measurements represented by positive-operator-va!ued measures (POVM) have been applied to extend the proof to two dimensional quantum systems. This note gives numerical calculations on the first KS proof for a single qubit.
Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the noncontextual hidden variable theory of our nature at the microscopic scale. The conflict between the noncontextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality inequalities in a state-(in)dependent manner. Here we propose a systematic and programmable construction of a state-independent proof from a given set of nonspecific rays in C3{\mathbb{C}}^{3}C3 according to their Gram matrix. This approach brings us a greater convenience in the experimental arrangements. Besides, our proofs in C3{\mathbb{C}}^{3}C3 can also be generalized to any higher-dimensional systems by a recursive method.