Partial separability revisited: Necessary and sufficient criteria

TL;DR

This paper extends the classification of mixed quantum states into partial separability classes for systems of arbitrary size, providing complete criteria for class membership, especially for tripartite systems, using convex roof extensions and algebraic methods.

Contribution

It introduces a comprehensive classification of mixed states with 20 classes for tripartite systems and establishes necessary and sufficient criteria for these classes, including new algebraic approaches.

Findings

Complete classification of mixed states into 20 partial separability classes.

Derived necessary and sufficient criteria for classifying states.

Applied convex roof extensions and algebraic methods for criteria derivation.

Abstract

We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.