A "quantum" Ramsey theorem for operator systems

TL;DR

This paper establishes a quantum analogue of Ramsey's theorem for operator systems, demonstrating that large enough matrix subspaces contain specific projections with predictable dimensional properties.

Contribution

It introduces a quantum Ramsey theorem for operator systems, extending classical combinatorial results into the quantum operator algebra setting.

Findings

Existence of a rank k projection with specific dimensional properties in large operator systems

Extension of classical Ramsey theory to quantum operator systems

Provides bounds for the size of operator systems containing certain projections

Abstract

Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under the formation of Hermitian adjoints. We prove that if n is sufficiently large then there exists a rank k orthogonal projection P such that dim(PVP) = 1 or k^2.