Math Graduate Student Seminar

A \emph{concrete operator system} is a vector space of bounded linear operators on a (finite-dimensional) Hilbert space. Recently, an interpretation of these objects in terms of zero-error quantum information theory has spurred interest in a combinatorial approach, the so-called ``noncommutative graph theory''. In late 2015, Nik Weaver proved his quantum Ramsey theorem, which says that for any concrete operator system, there must be either a large subspace on which the action of the operator system is trivial or one on which the action is isomorphic to a full matrix subalgebra. In surprising contrast to the classical case, the bound on the subspace dimension is related polyonimally to the dimension of the hilbert space. We'll discuss how this can be used to give some tight control on the combinatorics of noncommutative graphs. Our main result is a probabilistic method argument showing a partial converse (a lower bound on the "quantum Ramsey number") which is asymptotically tight up to logarithmic factors in the off-diagonal regime. Along the way, we'll introduce tools from random matrix theory. This is joint work with Martino Lupini, Matthew Kennedy, Mart\'\in Argerami, and Marcin Sabok.