Large violations in Kochen Specker contextuality and their applications

Abstract

The Kochen-Specker (KS) theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen-Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension $d\ge 2^{17}$ with the ratio of quantum value to classical bias being $O(\sqrt{d}/\log d)$. We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen-Specker proofs always consist of substructures of state-dependent contextuality proofs called $01$-gadgets or bugs. We show a one-to-one connection between $01$-gadgets in ${\mathbb{C}}^d$ and Hardy paradoxes for the maximally entangled state in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$. We use this connection to construct large violation $01$-gadgets between arbitrary vectors in ${\mathbb{C}}^d$, as well as novel Hardy paradoxes for the maximally entangled state in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$, and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in ${\mathbb{R}}^d$ is not a graph monotone, a result that that may be of independent interest.

16 pages, 4 figures

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• A simple test for hidden variables in spin-1 system
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• Strong Kochen-Specker theorem and incomputability of quantum randomness
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Cited by in corpus (2)

• Tilted Hardy paradoxes for device-independent randomness extraction
• Optimal Measurement Structures for Contextuality Applications