Measure of entanglement and the monogamy relation: a topical review

TL;DR

This topical review comprehensively surveys measures of quantum entanglement and their distribution across multipartite systems, with a focus on finite-dimensional settings. It systematizes the landscape by outlining bipartite measures and their monotones, majorization criteria, and the convex-roof framework, then extends to multipartite entanglement through k-entanglement, partitewise notions, and complete/multipartite measures. A central theme is the monogamy vs polygamy of entanglement, including refined (tight) monogamy relations, complete monogamy 2.0, and their implications for additivity, maximally entangled states, and information-theoretic tasks. The review also introduces a unifying framework for complete multipartite entanglement measures (MEMs) and complete GEMs, detailing how coarsening and hierarchy relations shape the distribution of quantum correlations and identifying open problems in higher dimensions and general mixed states.

Abstract

Characterizing entanglement, including quantifying and distribution of entanglement, which lies at heart of the quantum resource theory, have been investigated extensively ever since Bennett \etal proposed three seminal measures of entanglement in 1996. Up to now, there are numerous measures of entanglement that have been proposed from different point of view and plenty of monogamy relations have been explored which make the distribution of entanglement became more and more clear. While this is relatively easy in the case of pure states, it is much more intricate for the case of mixed quantum states especially with higher dimension and more particles in the system. We present here an overview of the theory along this line. We outline most of the results in this field historically and focus on the finite-dimensional systems. In particular we emphasize the point of view that (i) which yardsticks haven been applied in quantifying entanglement and its distribution, (ii) what are the substantive characteristics and interrelations of these measures and their monogamy relations mathematically by comparing, and (iii) which concepts should be improved or revised and how they were developed accordingly.

Figures (15)

• Figure 1: In a standard quantum communication setting two parties Alice and Bob may perform any generalized measurement that is localized to their laboratory and communicate classically. The brick wall indicates that no quantum particles may be exchanged coherently between Alice and Bob. This set of operations is generally referred to as LOCC.
• Figure 2: Schematic picture of the action of quantum operations with and without sub-selection shown in part (a) and part (b) respectively.
• Figure 3: Schematic picture of the $k$-separable states.
• Figure 4: (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.
• Figure 5: Schematic picture of the sets of MEMs, U-g-MEMs, and C-g-MEMs.

Theorems & Definitions (7)

• Definition 1: GG2018q
• Definition 2: Jin2022aqt
• Theorem 1: Monogamy criterion in GG2019pra
• Definition 3
• Definition 4
• Definition 5
• Definition 6