The "quantum" Turan problem for operator systems

TL;DR

This paper investigates the quantum analogs of the Turan problem within operator systems, establishing bounds on the dimensions of subspaces that guarantee the existence of quantum cliques and anticliques.

Contribution

It introduces bounds on the dimensions of operator systems that ensure the presence of quantum cliques and anticliques, extending classical combinatorial concepts to quantum settings.

Findings

Derived bounds for the dimension of operator systems guaranteeing quantum k-anticliques.

Derived bounds for the dimension of operator systems guaranteeing quantum k-cliques.

Provided theoretical framework connecting quantum cliques and anticliques with operator system dimensions.

Abstract

Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under Hermitian transpose. A "quantum k-clique" for V is a rank k orthogonal projection P in M_n(C) for which dim(PVP) = k^2, and a "quantum k-anticlique" is a rank k orthogonal projection for which dim(PVP) = 1. We give upper and lower bounds both for the largest dimension of V which would ensure the existence of a quantum k-anticlique, and for the smallest dimension of V which would ensure the existence of a quantum k-clique.