Analytically computable tangle for three-qubit mixed states

TL;DR

This paper introduces the r-tangle, a new analytical tripartite entanglement measure for three-qubit mixed states, which simplifies the analysis of SLOCC-equivalent states and extends the concept of concurrence.

Contribution

The paper proposes the r-tangle as a novel, analytically computable entanglement measure for three-qubit mixed states, with advantages over the traditional tangle.

Findings

Derived an analytical form of the r-tangle for mixtures of generalized GHZ and W states.

Showed the r-tangle's analytical form applies to all SLOCC-equivalent states.

Established the r-tangle as a suitable three-partite counterpart to concurrence.

Abstract

We present a new tripartite entanglement measure for three-qubit mixed states. The new measure $t_{\mathrm{r}}(\rho)$, which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(\rho)$ for a three-qubit mixed state $\rho$, we can also derive $t_{\mathrm{r}}(\rho')$ analytically for any states $\rho'$ which are SLOCC-equivalent to the state $\rho$. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle $t_{\mathrm{r}}$ is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle $t_{\mathrm{r}}$ for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.