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arXiv:2112.10509 · Physical Review Research

Geometric mean of bipartite concurrences as a genuine multipartite entanglement measure

Yinfei Li, Jiangwei Shang

Published Apr 20, 2022• 42 citations
arXivHTMLPDF

Abstract

In this paper, we propose the geometric mean of bipartite concurrences as a genuine multipartite entanglement measure. This measure achieves the maximum value for absolutely maximally entangled states and has desirable properties for quantifying potential quantum resources. The simplicity and symmetry in the definition facilitates its computation for various multipartite entangled states including the GHZ states and the $W$ states. With explicit examples, we show that our measure results in distinct entanglement orderings from other measures and can detect differences in certain types of genuine multipartite entanglement while other measures cannot. These results justify the potential application of our measure for tasks involving genuine multipartite entanglement.

DOI
10.1103/physrevresearch.4.023059

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Multipartite entanglement is a fundamental aspect of quantum mechanics, crucial to advancements in quantum information processing and quantum computation. Within this field, Genuinely Multipartite Entanglement (GME), being entangled in all bipartitions, and Absolutely Maximally Entanglement (AME), maximally entangled in all bipartitions, represent two significant types of entanglement with diverse applications. In this work, we introduce a new measure called the GME-AME multipartite entanglement measure, with a non-zero value representing the GME states and the maximum value is reached only by the AME states. The measure is applied to study the multipartite entanglement of four partite systems using the operator to state mapping, and the four partite permutation qutrit states are classified according to the measure. With various examples, we show that our measure is robust in classifying the four partite entangled states.
ABSTRACT Characterizing the entanglement in multipartite quantum systems remains a central challenge in the theory of quantum entanglement. We establish a comprehensive framework for constructing ‐entanglement measures by leveraging bipartite entanglement measures and general ‐entanglement measures () with arithmetic and geometric averaging schemes. Numerical examples demonstrate that these measures outperform the existing ones, such as ‐ME concurrence, geometric mean (k‐GM) concurrence, and the generalized ‐entanglement measure by inducing distinct entanglement orderings and offering superior resolution. Furthermore, we reveal the operational relationship between the ‐entanglement and the local quantum coherence, verifying that the persistence of entanglement is intrinsically constrained by the system's capacity to maintain the local coherence. Finally, to overcome the inherent limitation of the conventional three‐tangle in characterizing the genuine multipartite entanglement of the GHZ‐W mixed states, we design a novel GME measure. Numerical results confirm that this measure effectively eliminates the detection blind spots where vanishes, providing a complete and practical mathematical tool for the quantitative analysis of genuine multipartite quantum correlations.
We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of N-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using the geometric mean area of these concurrence triangles, which are non-increasing under local operation and classical communication. The GME measures classify which parts are separable or entangled with the rest ones for non genuine entangled pure states. The GME measures for mixed states are given via the convex roof construction, and a witness to detect the GME of multipartite mixed states is presented by an approach based on state purifications. Detailed examples are given to illustrate the effectiveness of our GME measures.
Abstract A genuine multipartite entanglement measure based on the geometric method is investigated in this paper. This measure has desirable properties for quantifying the genuine multipartite entanglement. A lower bound of the genuine multipartite entanglement measure derived with the fidelity‐based method is then presented. The advantages of the measure proposed here with other measures are also presented. At last, examples are presented to show that the genuine entanglement measure has distinct entanglement ordering from other measures.