Complete $k$-partite entanglement measure

TL;DR

This work defines a rigorous framework for complete $k$-partite entanglement measures by leveraging coarsening relations among partitions. It establishes axioms for unified and complete $k$-PEMs and presents two construction schemes: one based on a unified multipartite entanglement measure and another on a minimal-sum approach, with completeness tied to subadditivity of the underlying reduced function. The authors show that one class yields complete $k$-PEMs under subadditivity, while the other is unified (and complete for $k=2$) but generally not complete for larger $k$, and they illustrate these results with explicit examples. The findings advance consistent quantification of multipartite entanglement across arbitrary partitions and hold potential for extending to other multipartite quantum correlations and monogamy relations.

Abstract

The $k$-partite entanglement, which focus on at most how many particles in the global system are entangled but separable from other particles, is complementary to the $k$-entanglement that reflects how many splitted subsystems are entangled under partitions of the systems in characterizing multipartite entanglement. Very recently, the theory of the complete $k$-entanglement measure has been established in [Phys. Rev. A 110, 012405 (2024)]. Here we investigate whether we can define the complete measure of the $k$-partite entanglement. Consequently, with the same spirit as that of the complete $k$-entanglement measure, we present the axiomatic postulates that a complete $k$-partite entanglement measure should require. Furthermore, we present two classes of $k$-partite entanglement measures and show that one is complete while the other one is unified but not complete except for the case of $k=2$.

Figures (1)

• Figure 1: (color online). (a) $3$-partite entangled pure state $|\Psi\rangle=|\psi\rangle^{A'_1A'_2A'_3}|\psi\rangle^{B'_1B'_2\cdots B'_p} \cdots|\psi\rangle^{X'_1X'_2\cdots X'_q}|\phi\rangle^{A_1A_2}|\phi\rangle^{B_1B_2}\cdots|\phi\rangle^{X_1X_2}|\varphi\rangle^A|\varphi\rangle^B\cdots$$|\varphi\rangle^X$, where $|\psi\rangle^{A'_1A'_2A'_3}$, $|\psi\rangle^{B'_1B'_2\cdots B'_p}$, $\dots$, $|\psi\rangle^{X'_1X'_2\cdots X'_q}$ are genuinely entangled states, $3\leqslant p\leqslant q$, $|\phi\rangle^{A_1A_2}$, $|\phi\rangle^{B_1B_2}$, $\dots$, $|\phi\rangle^{X_1X_2}$ are entangled states. In fact, if one of $|\psi\rangle^{A'_1A'_2A'_3}$, $|\psi\rangle^{B'_1B'_2\cdots B'_p}$, $\dots$, $|\psi\rangle^{X'_1X'_2\cdots X'_q}$ is genuinely entangled, $|\Psi\rangle$ is also $3$-partite entangled. Here we just take the general form of a $3$-partite entangled pure state. (b) $|\Phi\rangle=|\psi\rangle^{A_1A_2\cdots A_k}|\psi\rangle^{B_1B_2\cdots B_l}$ with $k$, $l\geqslant 0$, $k+l\geqslant3$, is a $3$-entangled pure state if one of the following is true: (i) $|\psi\rangle^{A_1A_2\cdots A_k}$ and $|\psi\rangle^{B_1B_2\cdots B_l}$ are genuinely entangled states, $k$, $l\geqslant3$, (ii) $|\psi\rangle^{A_1A_2\cdots A_k}$ and $|\psi\rangle^{B_1B_2\cdots B_l}$ are entangled states, $k=l=2$, (iii) If $k=0$ or $l=0$, $|\Phi\rangle$ is genuinely entangled.

Theorems & Definitions (5)

• theorem 1
• proof
• theorem 2
• theorem 3
• proof