Entanglement marginal problems

TL;DR

This paper introduces hierarchies of semidefinite programming relaxations to determine if reduced quantum states are compatible with a global separable state, connecting quantum marginal problems to classical analogs and providing efficient approximations.

Contribution

It develops new hierarchies of relaxations for the entanglement marginal problem and links their completeness to classical marginal problems, enabling efficient approximations in various quantum systems.

Findings

Hierarchies can approximate the set of marginals with arbitrary accuracy.

Complexity is polynomial in system size for certain configurations.

Results extend to infinite and higher-dimensional systems with symmetries.

Abstract

We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.