$k$-Entanglement Measure for Multipartite Systems without Convex-Roof Extensions and its Evaluation
Authors
Jie Guo
Shuyuan Yang
Jinchuan Hou
Xiaofei Qi
Kan He
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.
