A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios

TL;DR

This paper introduces a hierarchy of semidefinite programs to determine whether a given EPR scenario assemblage is quantum-realisable, improving the ability to identify non-quantum assemblages and bounding quantum advantages in information tasks.

Contribution

It develops a new hierarchy of semidefinite programs that converges to a set containing all quantum assemblages, aiding in the analysis of EPR correlations.

Findings

Hierarchy can determine non-membership of quantum assemblages

Hierarchy converges to a set containing all quantum assemblages

Enables upper bounds on quantum violations and advantages

Abstract

Correlations in Einstein-Podolsky-Rosen (EPR) scenarios, captured by \textit{assemblages} of unnormalised quantum states, have recently caught the attention of the community, both from a foundational and an information-theoretic perspective. The set of quantum-realisable assemblages, or abbreviated to quantum assemblages, are those that arise from multiple parties performing local measurements on a shared quantum system. In general, deciding whether or not a given assemblage is a quantum assemblage, i.e. membership of the set of quantum assemblages, is a hard problem, and not always solvable. In this paper we introduce a hierarchy of tests where each level either determines non-membership of the set of quantum assemblages or is inconclusive. The higher the level of the hierarchy the better one can determine non-membership, and this hierarchy converges to a particular set of assemblages. Furthermore, this set to which it converges contains the quantum assemblages. Each test in the hierarchy is formulated as a semidefinite program. This hierarchy allows one to upper bound the quantum violation of a steering inequality and the quantum advantage provided by quantum EPR assemblages in a communication or information-processing task.