On an Extension Problem for Density Matrices

TL;DR

This paper explores the complex problem of determining the existence of a three-part quantum density matrix with specified two-party marginals, providing partial necessary and sufficient conditions and highlighting differences from classical cases.

Contribution

It offers new partial results with conditions for the existence of a global density matrix given certain marginals, advancing understanding of quantum marginal problems.

Findings

Identifies necessary conditions for the existence of the density matrix.

Provides sufficient conditions for the density matrix existence.

Highlights differences between quantum and classical marginal problems.

Abstract

We investigate the problem of the existence of a density matrix rho on the product of three Hilbert spaces with given marginals on the pair (1,2) and the pair (2,3). While we do not solve this problem completely we offer partial results in the form of some necessary and some sufficient conditions on the two marginals. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, trace_1 (rho_{12}) = \trace_3 (rho_{23}).