The “quantum” Turán problem for operator systems

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 $d i m (P V P) = k^{2}$ , and a “quantum $k$ -anticlique” is a rank $k$ orthogonal projection for which $d i m (P V P) = 1$ . 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.

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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

Authors

Nik Weaver
Department of Mathematics Washington University Saint Louis, MO United States

Vol. 301, No. 1, 2019

Nik Weaver

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors