Deterministic LOCC transformation of three-qubit pure states and entanglement transfer

TL;DR

This paper establishes a complete criterion for deterministic LOCC transformations among three-qubit pure states, revealing a partial order based on five known and one new entanglement parameters, and clarifies entanglement transfer rules.

Contribution

It introduces a necessary and sufficient condition for LOCC transformations of three-qubit states, extending Nielsen's work with a new entanglement parameter and detailed transfer rules.

Findings

Defines a partial order of three-qubit states via five entanglement parameters and a new parameter.

Provides a criterion for deterministic LOCC transformations between three-qubit states.

Determines the minimum number of measurements needed for arbitrary transformations.

Abstract

A necessary and sufficient condition of the possibility of a deterministic local operations and classical communication (LOCC) transformation of three-qubit pure states is given. The condition shows that the three-qubit pure states are a partially ordered set parametrized by five well-known entanglement parameters and a novel parameter; the five are the concurrences $C_{AB}$, $C_{AC}$, $C_{BC}$, the tangle $\tau_{ABC}$ and the fifth parameter $J_{5}$ of Ref. [18], while the other new one is the entanglement charge $Q_{e}$. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen's work [20] to three-qubit pure states. We also clarify the rules of transfer and dissipation of the entanglement which is caused by deterministic LOCC transformations. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation between three-qubit pure states is given.