The additivity of states uniquely determined by marginals

TL;DR

This paper extends the concept of uniquely determined states by marginals from pure to mixed states, explores the additivity property under tensor products, and proposes methods to construct genuinely multipartite entangled states that are uniquely determined by their marginals.

Contribution

It generalizes the notion of UDA states to mixed states, investigates additivity properties under tensor products, and introduces a way to construct GME states that are also UDA.

Findings

Additivity holds if one state is pure.

Conditions for additivity with two mixed UDA states.

Method to construct GME states that are also UDA.

Abstract

The pure states that can be uniquely determined among all (UDA) states by their marginals are essential to efficient quantum state tomography. We generalize the UDA states from the context of pure states to that of arbitrary (no matter pure or mixed) states, motivated by the efficient state tomography of low-rank states. We call the \emph{additivity} of $k$-UDA states for three different composite ways of tensor product, if the composite state of two $k$-UDA states is still uniquely determined by the $k$-partite marginals for the corresponding type of tensor product. We show that the additivity holds if one of the two initial states is pure, and present the conditions under which the additivity holds for two mixed UDA states. One of the three composite ways of tensor product is also adopted to construct genuinely multipartite entangled (GME) states. Therefore, it is effective to construct multipartite $k$-UDA state with genuine entanglement by uniting the additivity of $k$-UDA states and the construction of GME states.