QLunch: Máté Farkas

Speaker: Máté Farkas, University of York

Title: Semidefinite representability of the set of quantum correlations in the simplest Bell scenario

Abstract: In this talk I will look at the simplest Bell scenario: two parties both with two inputs and two outputs. It is well-known that in this setting the extremal points of the set of quantum correlations can be achieved by locally two-dimensional systems. This simplification makes it possible to cast the problem of maximising a linear functional (Bell inequality) over this set as a polynomial optimisation problem. Using results from polynomial optimisation, we show that for every Bell inequality there exists a finite  semidefinite programme that finds its maximum. Through a connection with entanglement theory, we also show that there is a uniform bound on the size of this semidefinite programme for every Bell inequality. I will further discuss the semidefinite representability of this set (whether the membership problem is a finite semidefinite programme) and whether the well-known Navascués--Pironio--Acín hierarchy of semidefinite programmes converges on a finite level for optimising Bell inequalities in the simplest Bell scenario. Both of these questions remain open.