Extending Schmidt vector from pure to mixed states for characterizing entanglement

TL;DR

This paper extends the concept of Schmidt vector from pure to mixed bipartite states, providing a new tool for characterizing and quantifying entanglement through majorization and monotonicity properties.

Contribution

It introduces a novel definition of Schmidt vector for mixed states, demonstrating its properties and relation to entanglement measures, and extends Schmidt rank to mixed states.

Findings

Schmidt vector fully characterizes separable and maximally entangled states.

Schmidt vector is monotonic under LOCC, providing necessary conditions for state conversion.

Introduces entanglement monotones based on concave, symmetric functions of the Schmidt vector.

Abstract

In this study, we enhance the understanding of entanglement transformations and their quantification by extending the concept of Schmidt vector from pure to mixed bipartite states, exploiting the lattice structure of majorization. The Schmidt vector of a bipartite mixed state is defined using two distinct methods: as a concave roof extension of Schmidt vectors of pure states, or equivalently, from the set of pure states that can be transformed into the mixed state through local operations and classical communication (LOCC). We demonstrate that the Schmidt vector fully characterizes separable and maximally entangled states. Furthermore, we prove that the Schmidt vector is monotonic and strongly monotonic under LOCC, giving necessary conditions for conversions between mixed states. Additionally, we extend the definition of the Schmidt rank from pure states to mixed states as the cardinality of the support of the Schmidt vector and show that it is equal to the Schmidt number introduced in previous work [Phys. Rev. A 61, 040301 (R), 2000]. Finally, we introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.