Entanglement measure for the W-class states

TL;DR

This paper investigates the structure and quantification of entanglement in W-class states, establishing conditions for separability and proposing effective measures like the sum of two-tangles, addressing limitations of the $ ext{pi}$-tangle.

Contribution

It introduces a new entanglement measure based on the sum of two-tangles and provides conditions linking separability to pairwise entanglement in W-class states.

Findings

Global separability is linked to the absence of pairwise entanglement.

The sum of two-tangles effectively quantifies entanglement in W-class states.

The sum of $ ext{pi}$-tangles addresses the limitations of the $ ext{pi}$-tangle for large systems.

Abstract

The structure and quantification of entanglement in the W-class states are investigated under physically motivated transformations that induce mixed-state dynamics. A rigorous condition is established linking global separability to the behavior of pairwise entanglement, showing that the absence of pairwise entanglement is sufficient to guarantee complete separability of the system, provided the Hilbert-space basis is preserved. This result motivates the identification of the sum of two-tangles as a natural and effective entanglement quantifier for the W-class states. Furthermore, the commonly used $\pi$-tangle becomes ineffective for the maximally entangled $n$-qubit W state as the system size increases, vanishing in the large-$n$ limit. To address this limitation, the sum of $\pi$-tangles is introduced, which, like the sum of two-tangles, successfully quantifies the entanglement of the maximally entangled $n$-qubit W state in the large-$n$ limit. In addition, a new condition for entanglement measures is introduced, which facilitates the formulation of a well-behaved and physically meaningful entanglement measure.