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

TL;DR

This work introduces a universal axiomatic framework for multipartite entanglement that defines a true $k$-entanglement measure $E_w^{(k,n)}$ without convex-roof extensions. Grounded in entanglement witnesses and a maximization over $k$-partitions through $g_n^{(k)}(L)$, the measure is faithful, LOCC-monotone, convex, and obeys unification and hierarchy, thereby treating $k$-entanglement as a bona fide resource. The authors provide a universal evaluation strategy and an open-source software suite that computes $E_w^{(k,n)}$ for $n$-partite finite-dimensional systems (and specifically for $n$-qubit cases with $2\leq n\leq 4$), demonstrating accuracy against known thresholds and improved precision on noisy states such as Werner and W states. This framework offers scalable, practical multipartite entanglement quantification with broad implications for quantum information processing and experimental verification of quantum resources.

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.

Figures (7)

• Figure 1: Entanglement Measure $\tilde{E}_{w,2}^{(2,2)}$ (red $-{\small +}-$) and the entanglement negativity $\mathcal{N}$ (green $-\cdot-$) of $2$-qubit Werner state and W state mixed with white noise.
• Figure 2: The $k$-E measures $\tilde{E}_{w,n}^{(k,n)}$ at the $n$-qubit Werner states $\varrho(p)$ for $n = 2, 3, 4$, as functions of the parameter $p$. The plot illustrates the thresholds detected by $\tilde{E}_{w,n}^{(k,n)}$ for $\varrho(p)$ to be $k$-E, highlighting the boundaries that distinguish $k$-separability from $k$-E.
• Figure 3: The $k$-E measures $\tilde{E}_{w,n}^{(k,n)}$ at the $n$-qubit W states mixed with white noise $\varrho(p)$ for $n = 2, 3, 4$, as functions of the parameter $p$. The plot illustrates the thresholds detected by $\tilde{E}_{w,n}^{(k,n)}$ for $\varrho(p)$ to be $k$-E, highlighting the boundaries that distinguish $k$-separability from $k$-E.
• Figure 4: The $k$-E measures $\tilde{E}_{w,4}^{(k,4)}$ for the 4-qubit $|{Cl}_{4}\rangle$ state mixed with white noise, as a function of the parameter $p$. The plot illustrates the transitions between $k$-separability and $k$-E classes.
• Figure 5: The $k$-E measures $\tilde{E}_{w,4}^{(k,4)}$ for the 4-qubit $|D_{24}\rangle$ state mixed with white noise, as a function of the parameter $p$.