Large violations in Kochen Specker contextuality and their applications

TL;DR

This paper introduces large violation non-contextuality inequalities based on Kochen-Specker proofs, explores their properties, and demonstrates applications including new Hardy paradoxes and insights into graph representations in quantum contextuality.

Contribution

It presents the first state-independent non-contextuality inequalities with large violations, establishes a connection between KS proofs and Hardy paradoxes, and investigates graph representation dimensions.

Findings

Quantum violation ratio scales as O(√d / log d) in high dimensions.

Constructs large violation 01-gadgets and Hardy paradoxes in high-dimensional spaces.

Shows the non-monotonicity of the minimum dimension of faithful orthogonal graph representations.

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.