Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

TL;DR

This paper develops a hierarchy of semidefinite programming bounds to analyze bipartite quantum correlations and entanglement dimensions, connecting quantum graph parameters with noncommutative polynomial optimization.

Contribution

It introduces a new hierarchy for lower bounds on entanglement dimensions and unifies quantum graph parameters within a tracial optimization framework.

Findings

Hierarchy converges to minimal average entanglement dimension.

Establishes correspondence between entanglement dimension and completely positive semidefinite rank.

Unifies bounds on quantum chromatic and stability numbers using tracial optimization.

Abstract

In this paper we study bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. For synchronous correlations, we show a correspondence between the minimal entanglement dimension and the completely positive semidefinite rank of an associated matrix. We then study optimization over the set of synchronous correlations by investigating quantum graph parameters. We unify existing bounds on the quantum chromatic number and the quantum stability number by placing them in the framework of tracial optimization. In particular, we show that the projective packing number, the projective rank, and the tracial rank arise naturally when considering tracial analogues of the Lasserre hierarchy for the stability and chromatic number of a graph. We also introduce semidefinite programming hierarchies converging to the commuting quantum chromatic number and commuting quantum stability number.