In this paper, we generalize the residual entanglement to the case of multipartite states in arbitrary dimensions by making use of a different method. Through the introduction of a special entanglement measure, the residual entanglement of mixed states takes on a form that is more elegant than that in Ref. [7] used by Coffman et al. [Phys. Rev. A 61, 052306 (2000)]. The result obtained in this paper is different from the previous one given in Ref. [8] by Wong and Christensen [Phys. Rev. A 63, 044301 (2000)]. Several examples demonstrate that our present result is a good measurement of the multipartite entanglement. Furthermore, the original residual entanglement is a special case of our result.
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≤k≤n, we construct a witness-based true
k-entanglement measure
Ew(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
Ew(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
Ew(k,n) for systems up to four qubits. Numerical tests show that the software
is sufficiently accurate and reliable.
We introduce a unified, computable measure of multipartite entanglement based on the spectral properties of an entanglement graph and its associated entanglement matrix. This framework quantifies quantum correlations among arbitrary subsystems and partitions of a composite system. We prove that the resulting spectral entanglement measure satisfies the fundamental requirements of entanglement measures. Furthermore, we derive a generic multipartite monogamy relation that extends residual entanglement beyond qubit systems and introduces spectral residual entanglement for arbitrary multipartite states.