Large violations in Kochen Specker contextuality and their applications

• URL: http://arxiv.org/abs/2106.15954v2
• Date: Thu, 1 Jul 2021 17:45:45 GMT
• Title: Large violations in Kochen Specker contextuality and their applications
• Authors: Ravishankar Ramanathan, Yuan Liu, Pawe{\l} Horodecki
• Abstract summary: We show Kochen-Specker proofs in Hilbert spaces of dimension $d geq 217$ with the ratio of quantum value to classical bias being $O(sqrtd/log d)$. We show a one-to-one connection between $01$-gadgets in $mathbbCd$ and Hardy paradoxes for the maximally entangled state in $mathbbCd otimes mathbbCd$.
• Score: 4.398712854913047
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $d \geq 2^{17}$ 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.

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