On the relation between Schmidt coefficients and entanglement

TL;DR

This paper explores how Schmidt coefficients relate to entanglement in bipartite quantum states, introducing Schmidt equivalence classes and new criteria for separability based on these coefficients.

Contribution

It introduces the concept of Schmidt equivalence classes and proposes new separability criteria based on symmetric polynomials of Schmidt coefficients.

Findings

Schmidt coefficients determine entanglement properties.

Realignment criterion depends only on Schmidt equivalence class.

New criteria generalize existing separability tests.

Abstract

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the `realignment' or `computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance -- with regard to the characterization of entanglement -- of the `symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.