An introduction to entanglement measures

TL;DR

The paper surveys the theory of entanglement measures with a focus on bipartite, finite- and infinite-dimensional systems under LOCC. It develops an axiomatic framework for entanglement measures and reviews a broad set of quantitative tools, including E_C, E_D, E_F, relative entropy of entanglement, squashed entanglement, and log-negativity, highlighting their properties, interrelations, and computational aspects. The asymptotic regime is shown to yield a natural ordering for pure states (via entropy of entanglement) but reveals irreversibility for mixed states, motivating the search for universal measures and bounds. The discussion extends to continuous-variable and Gaussian settings, and to multipartite entanglement, where a complete theory remains elusive, leaving numerous open problems about additivity, distillability, and reversibility under broader operation classes.

Abstract

We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. Topics covered include: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; the distillable entanglement; the relative entropic measures; the squashed entanglement; log-negativity; the robustness monotones; the greatest cross-norm; uniqueness and extremality theorems. Infinite dimensional systems and multi-party settings will be discussed briefly.

Figures (6)

• 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 (eqs. (\ref{['eq1']}) and (\ref{['eq2']}) respectively) shown in part (a) and part (b) respectively.
• Figure 3: Schematic picture of the situation described by eq. (\ref{['subadd']}). The entanglement of formation of an arbitrary four particle state $|\psi\rangle$, with particles held by parties $A$ and $B$ is given is given on the left hand side of eq. (\ref{['subadd']}). The right hand side of eq. (\ref{['subadd']}) is the sum of the entanglement of formation of the states $\rho_1=tr_{A_2B_2}|\psi\rangle\langle\psi|$ and $\rho_2=tr_{A_1B_1}|\psi\rangle\langle\psi|$ obtained by tracing out the lower upper half of the system.
• Figure 4: A schematic picture of the convex roof construction in one dimension. The non-convex function $f(x)$ is given by the solid line. The dotted curve is a convex function smaller than $f$ and the convex roof, the largest convex function that is smaller than $f$, is drawn as a dashed curved (it coincides in large parts with $f$).
• Figure 5: The relative entropy of entanglement is defined as the smallest relative entropy distance from the state $\rho$ to states $\sigma$ taken from the set $X$. The set $X$ may be defined as the set of separable states, non-distillable states or any other set that is mapped onto itself by LOCC.