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Multipartite entanglement measures: a review

Mengru Ma, Yinfei Li, Jiangwei Shang

TL;DR

This review analyzes the quantification of multipartite entanglement by contrasting axiomatic (monotone) and operational measures. It surveys a broad suite of ME and GME metrics, including squashed entanglement, three-tangle, Schmidt measure, GM/GGM, GME-concurrence, and teleportation-based criteria, and connects these measures to tasks such as distillation, private keys, and teleportation fidelity. The authors discuss properties, limitations, and hierarchy among measures, and outline operational notions like accessible/source entanglement and concentratable entanglement to capture practical utility under LOCC. They also highlight open questions—most notably universal GME quantification and extensions to mixed states—and argue that advancing these measures will impact quantum information processing in multipartite settings. Overall, the review provides a comprehensive roadmap for characterizing and leveraging genuine multipartite entanglement in quantum technologies.

Abstract

Quantum entanglement, a fundamental aspect of quantum mechanics, has captured significant attention in the era of quantum information science. In multipartite quantum systems, entanglement plays a crucial role in facilitating various quantum information processing tasks, such as quantum teleportation and dense coding. In this article, we review the theory of multipartite entanglement measures, with a particular focus on the genuine as well as the operational meaning of multipartite entanglement measures. By providing a thorough and valuable insight on this field, we hope that this review would inspire and guide researchers in their endeavors to further develop novel approaches for characterizing multipartite entanglement.

Multipartite entanglement measures: a review

TL;DR

This review analyzes the quantification of multipartite entanglement by contrasting axiomatic (monotone) and operational measures. It surveys a broad suite of ME and GME metrics, including squashed entanglement, three-tangle, Schmidt measure, GM/GGM, GME-concurrence, and teleportation-based criteria, and connects these measures to tasks such as distillation, private keys, and teleportation fidelity. The authors discuss properties, limitations, and hierarchy among measures, and outline operational notions like accessible/source entanglement and concentratable entanglement to capture practical utility under LOCC. They also highlight open questions—most notably universal GME quantification and extensions to mixed states—and argue that advancing these measures will impact quantum information processing in multipartite settings. Overall, the review provides a comprehensive roadmap for characterizing and leveraging genuine multipartite entanglement in quantum technologies.

Abstract

Quantum entanglement, a fundamental aspect of quantum mechanics, has captured significant attention in the era of quantum information science. In multipartite quantum systems, entanglement plays a crucial role in facilitating various quantum information processing tasks, such as quantum teleportation and dense coding. In this article, we review the theory of multipartite entanglement measures, with a particular focus on the genuine as well as the operational meaning of multipartite entanglement measures. By providing a thorough and valuable insight on this field, we hope that this review would inspire and guide researchers in their endeavors to further develop novel approaches for characterizing multipartite entanglement.
Paper Structure (31 sections, 63 equations, 2 figures)

This paper contains 31 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of entanglement measures
  3. The LOCC paradigm
  4. Entanglement monotonicity
  5. Postulates of entanglement measures
  6. Convex-roof extension
  7. Entanglement of formation
  8. Concurrence
  9. Robustness of entanglement
  10. ME measures
  11. Squashed entanglement
  12. Three-tangle
  13. Global entanglement
  14. Schmidt measure
  15. Complete ME measure
  16. ...and 16 more sections

Figures (2)

  • Figure 1: An illustration of the SLOCC classes for three-qubit states PhysRevLett.87.040401. S : the fully-separable class; B: the biseparable class, which is the convex hull of biseparable states w.r.t. an arbitrary partition; $W$: the $W$ class; GHZ: the GHZ class.
  • Figure 2: Schematic representation of the set of all states for bipartite systems, including the SEP states, PPT entangled states, hypothetical NPT nondistillable states, and distillable states.