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arXiv:quant-ph/9412006 · Physics Letters A

Kochen-Specker theorem for eight-dimensional space

Michael William Kernaghan, Asher Peres

Published Feb 1, 1995• 85 citations
arXivHTMLPDF
DOI
10.1016/0375-9601(95)00012-r

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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.
For eight-dimensional quantum systems there is a Kochen-Specker (KS) set of 40 quantum yes-no tests that is related to the Greenberger-Horne-Zeilinger (GHZ) proof of Bell's theorem. Here we experimentally implement this KS set using an eight-dimensional Hilbert space spanned by the transverse momentum of single photons. We show that the experimental results of these tests violate a state-independent noncontextuality inequality. In addition, we show that, if the system is prepared in states that are formally equivalent to a three-qubit GHZ and W states, then the results of a subset of 16 tests violate a noncontextuality inequality that is formally equivalent to the three-party Mermin's Bell inequality, but for single eight-dimensional quantum systems. These experimental results highlight the connection between quantum contextuality and nonlocality for eight-dimensional quantum systems.
The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real eight-dimensional Hilbert space that contains a large number of parity proofs of the Kochen–Specker (KS) theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their KS diagram in the form of a 'basis table' showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the KS theorem found to date in eight-dimensions.