A complete hierarchy for the pure state marginal problem in quantum mechanics

TL;DR

This paper establishes a comprehensive framework for solving the pure state quantum marginal problem by linking it to the separability problem and providing a sequence of semidefinite programs for compatibility checks.

Contribution

It introduces a complete hierarchy of semidefinite programs to determine pure state marginals, connecting the problem to separability and quantum code existence.

Findings

Decides pure state marginal compatibility using semidefinite programming

Shows equivalence between maximally entangled states and separability of certain states

Relates quantum code existence to the marginal problem

Abstract

Clarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are compatible with some pure global quantum state. As an application, we prove that the existence of multiparticle absolutely maximally entangled states for a given dimension is equivalent to the separability of an explicitly given two-party quantum state. Finally, we show that the existence of quantum codes with given parameters can also be interpreted as a marginal problem, hence, our complete hierarchy can also be used.