Vol. 301, No. 1, 2019

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The “quantum” Turán problem for operator systems

Nik Weaver

Vol. 301 (2019), No. 1, 335–349
DOI: 10.2140/pjm.2019.301.335

Let V be a linear subspace of Mn() which contains the identity matrix and is stable under Hermitian transpose. A “quantum k-clique” for V is a rank k orthogonal projection P Mn() for which dim(PVP) = k2, 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.

operator systems, Turán problem, quantum graph theory
Mathematical Subject Classification 2010
Primary: 05C69, 05D10, 46L07, 81P45
Received: 20 April 2018
Accepted: 29 October 2018
Published: 16 September 2019
Nik Weaver
Department of Mathematics
Washington University
Saint Louis, MO
United States