The “quantum” Turán problem for operator systems

Nik Weaver

Vol. 301 (2019), No. 1, 335–349

DOI: 10.2140/pjm.2019.301.335

Abstract

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

Keywords

operator systems, Turán problem, quantum graph theory

Mathematical Subject Classification 2010

Primary: 05C69, 05D10, 46L07, 81P45

Milestones

Received: 20 April 2018

Accepted: 29 October 2018

Published: 16 September 2019