Multipartite entanglement measures

The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. Kökényesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some “unified” notion of entanglement.

Figure 1

Lattices of the labels of the first and second kinds and class labels, $P_{\text{I}},P_{\text{II}}$, and $P_{\text{III}}$, are illustrated for $n=2$. The partitions $\alpha \in P_{\text{I}}$ are denoted by small pictograms; the labels of the second kind $\mathbf{\alpha }\in P_{\text{II}}$ are down-sets of partitions (only the maximal elements are drawn). The class labels $\underline{\mathbf{\alpha }}\in P_{\text{III}}$ are up-sets of labels of the second kind (only the minimal elements are drawn). The order relation is denoted with an arrow: $\beta \to \alpha$ means $\beta ⪯\alpha ,\mathbf{\beta }\to \mathbf{\alpha }$ means $\mathbf{\beta }⪯\mathbf{\alpha }$, and $\underline{\mathbf{\beta }}\to \underline{\mathbf{\alpha }}$ means $\underline{\mathbf{\beta }}⪯\underline{\mathbf{\alpha }}$. By means of (16) and (32), the $P_{\text{I}}$ and $P_{\text{II}}$ resemble the inclusion of the sets of $\alpha$-separable pure (${\mathcal{P}}_{\alpha }$), $\alpha$-separable mixed (${\mathcal{D}}_{\alpha }$), and $\mathbf{\alpha }$-separable pure (${\mathcal{P}}_{\mathbf{\alpha }}$), $\mathbf{\alpha }$-separable mixed states (${\mathcal{D}}_{\mathbf{\alpha }}$) (see Secs. 3b and 3d). The lattice $P_{\text{III}}$ is the class hierarchy (see Sec. 3f), and by means of (54), it is related to the LOCC convertibility of the classes (${\mathcal{C}}_{\underline{\mathbf{\alpha }}}$).

Figure 2

Lattices of the labels of the first and second kinds and class labels, $P_{\text{I}},P_{\text{II}}$, and $P_{\text{III}}$, are illustrated for $n=3$. The partitions $\alpha \in P_{\text{I}}$ are denoted by small pictograms; the labels of the second kind $\mathbf{\alpha }\in P_{\text{II}}$ are down-sets of partitions, in which case the different elements are drawn with different colors (only the maximal elements are drawn). The class labels $\underline{\mathbf{\alpha }}\in P_{\text{III}}$ are up-sets of labels of the second kind; these are written side by side (only the minimal elements are drawn). The order relation is denoted with an arrow: $\beta \to \alpha$ means $\beta ⪯\alpha ,\mathbf{\beta }\to \mathbf{\alpha }$ means $\mathbf{\beta }⪯\mathbf{\alpha }$, and $\underline{\mathbf{\beta }}\to \underline{\mathbf{\alpha }}$ means $\underline{\mathbf{\beta }}⪯\underline{\mathbf{\alpha }}$. By means of (16) and (32), the $P_{\text{I}}$ and $P_{\text{II}}$ resemble the inclusion of the sets of $\alpha$-separable pure (${\mathcal{P}}_{\alpha }$), $\alpha$-separable mixed (${\mathcal{D}}_{\alpha }$), and $\mathbf{\alpha }$-separable pure (${\mathcal{P}}_{\mathbf{\alpha }}$), $\mathbf{\alpha }$-separable mixed states (${\mathcal{D}}_{\mathbf{\alpha }}$); (see in Secs. 3b and 3d). The lattice $P_{\text{III}}$ is the class hierarchy (see Sec. 3f), and by means of (54), it is related to the LOCC convertibility of the classes (${\mathcal{C}}_{\underline{\mathbf{\alpha }}}$).