SLOCC Convertibility between Two-Qubit States

TL;DR

This paper classifies certain four-qubit states and characterizes the set of separable states and SLOCC protocols for Bell-diagonal two-qubit states, providing a complete set of monotones and conditions for state convertibility.

Contribution

It introduces a classification of four-qubit states commuting with specific operators and derives a complete set of SLOCC monotones for Bell-diagonal states.

Findings

Complete set of SLOCC monotones for Bell-diagonal states

Necessary and sufficient conditions for two-qubit state convertibility

Characterization of separable states via entanglement witnesses

Abstract

In this paper we classify the four-qubit states that commute with $U\otimes{U}\otimes{V}\otimes{V}$, where $U$ and $V$ are arbitrary members of the Pauli group. We characterize the set of separable states for this class, in terms of a finite number of entanglement witnesses. Equivalently, we characterize the two-qubit, Bell-diagonal-preserving, completely positive maps that are separable. These separable completely positive maps correspond to protocols that can be implemented with stochastic local operations assisted by classical communication (SLOCC). This allows us to derive a complete set of SLOCC monotones for Bell-diagonal states, which, in turn, provides the necessary and sufficient conditions for converting one two-qubit state to another by SLOCC.