Entanglement measure for the W-class states

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 $π$-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 $π$-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.