Markovian Marignals

TL;DR

This paper introduces Markovian marginals, a framework for solving the quantum marginal problem by leveraging quantum Markov chain structures, with implications for states with finite correlation lengths.

Contribution

It presents a constructive approach to the quantum marginal problem using Markovian structures and connects it to physical states with finite correlation lengths.

Findings

Existence of a global state consistent with local marginals under Markovian structure.

Reduction of the marginal problem to a combinatorial problem.

Applicability to topologically ordered and finite temperature Gibbs states.

Abstract

We introduce a class of so called Markovian marginals, which gives a natural framework for constructing solutions to the quantum marginal problem. We consider a set of marginals that possess a certain internal quantum Markov chain structure. If they are equipped with such a structure and are locally consistent on their overlapping supports, there exists a global state that is consistent with all the marginals. The proof is constructive, and relies on a reduction of the marginal problem to a certain combinatorial problem. By employing an entanglement entropy scaling law, we give a physical argument that the requisite structure exists in any states with finite correlation lengths. This includes topologically ordered states as well as finite temperature Gibbs states.