Compatible quantum correlations: on extension problems for Werner and isotropic states

TL;DR

This paper explores the conditions under which certain bipartite quantum states, specifically Werner and isotropic states, can be extended to larger multipartite states, revealing the role of correlations beyond entanglement.

Contribution

It provides necessary and sufficient conditions for joinability and sharability of Werner and isotropic states, advancing understanding of quantum marginal problems with symmetry considerations.

Findings

Entanglement is necessary but not sufficient for joinability limitations.

Correlations beyond entanglement can restrict state joinability.

Explicit conditions for three-party joinability and 1-n sharability are derived.

Abstract

We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N-partite quantum state. Intuitively, we say that the multipartite state "joins" the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.