A $k$-entanglement measure complying with the many-body resource theory and its evaluation

Abstract:$k$-entanglement provides a hierarchical characterization of multipartite correlations according to their entanglement structure, ranging from multipartite entanglement ($k=n$) to genuine multipartite entanglement ($k=2$). Its quantitative treatment for mixed states faces three related obstacles: the standard axioms for such quantifications does not incorporate the requirements from many-body resource theory, as it does not regulate how to quantify the entanglement assigned to part systems of the same multipartite state and how to describe the rule that the entanglement possessed by a part system never exceeds the entanglement possed by whole system; no explicit measures are known that satisfy the above requirements from many-body resource theory; and the extreme difficulty in numerical evaluation at mixed states of known quantifications. In this work, we first formulate an axiomatic framework for \emph{true} $k$-entanglement measures that obeys the many-body resource theory, by supplementing usual faithfulness and LOCC monotonicity with two new ones: unification condition and hierarchy condition. Then, for every $2\leq k\leq n$, we construct a witness-based true $k$-entanglement measure $E_w^{(k,n)}$ without the convex-roof extension, which is valid for both finite- and infinite-dimensional $n$-partite systems. Finally, we derive a method of constructing finite-dimensional $k$-entanglement witnesses from positive operator and $k$-separable-state optimizations, design an evaluation scheme of $E_w^{(k,n)}$ based on pre-established database of $k$-entanglement witnesses sampled by positive operators. To check the feasibility of this computing scheme, we develop a software tool, which implements the computation of the approximation of $E_w^{(k,n)}$ for systems up to four qubits. Numerical tests show that the software is sufficiently accurate and reliable.

arXiv:2512.12588 [quant-ph]

(or arXiv:2512.12588v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2512.12588 Focus to learn more arXiv-issued DOI via DataCite

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