$k$-Entanglement Measure for Multipartite Systems without Convex-Roof Extensions and its Evaluation

TL;DR

This paper introduces a new, axiomatic $k$-entanglement measure for multipartite quantum systems that is computationally efficient, avoiding convex-roof extensions, and provides a practical tool for quantifying entanglement.

Contribution

The authors develop the first true $k$-entanglement measure satisfying all axioms, avoiding convex-roof extensions, and provide an efficient universal algorithm for its evaluation.

Findings

Successfully evaluated $k$-entanglement in four-qubit states within 200 seconds.

The measure aligns with necessary-and-sufficient criteria for entanglement detection.

Provides bounds and thresholds for multipartite entanglement, enhancing understanding and quantification.

Abstract

Multipartite entanglement underpins quantum technologies but its study is limited by the lack of universal measures, unified frameworks, and the intractability of convex-roof extensions. We establish an axiomatic framework and introduce the first \emph{true} $k$-entanglement measure, $E_w^{(k,n)}$, which satisfies all axioms, establishes $k$-entanglement as a multipartite quantum resource, avoids convex-roof constructions, and is efficiently computable. A universal algorithm evaluates arbitrary finite-dimensional states, with open-source software covering all partitions of four-qubit systems. Numerical tests certify $k$-entanglement within 200 seconds, consistent with necessary-and-sufficient criteria, tightening bounds and revealing new thresholds. This framework offers a scalable, practical tool for rigorous multipartite entanglement quantification.