Large violations in Kochen Specker contextuality and their applications

Abstract

It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. 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 ⩾ 217 with the ratio of quantum value to classical bias being $O(\sqrt{d}/\mathrm{log}\,d)$. 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. 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 may be of independent interest.