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arXiv:2108.03638 · Journal of Physics A Mathematical and Theoretical

Genuine multipartite entanglement measure

Yu Jing Guo, Yanping Jia, Xinping Li, Lizhong Huang

Published Feb 17, 2022• 41 citations
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Abstract

Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m -partite entanglement measures from any bipartite entanglement and any k -partite entanglement measure, 3 ⩽ k < m . In addition, as a complement to the three-qubit concurrence triangle proposed in ( Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.

DOI
10.1088/1751-8121/ac5649

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Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m -partite entanglement measures from any bipartite entanglement and any k -partite entanglement measure, 3 ⩽ k < m . In addition, as a complement to the three-qubit concurrence triangle proposed in ( Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.
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Genuine multipartite entanglement is crucial for quantum information and quantum technologies but quantifying it has been a long-standing challenge. Most existing measures do not meet a “genuine” requirement, making them unsuitable for many applications. Here, we present a surprising triangle measure for tripartite entanglement, and introduce the extension to four-qubit systems. We discuss potential avenues for further generalizations along this geometric path. This indicates that our work may pave the way for further advancements.
Genuine Multipartite Entanglement Measure Based on ααα-concurrence
Quantifying genuine entanglement is a crucial task in quantum information theory. Based on the geometric mean of bipartite ααα-concurrences among all bipartitions, we present a class of well-defined genuine multipartite entanglement (GME) measures GαααC with one parameter ααα for arbitrary multipartite states. We show that the GαααC is of continuity for any multipartite pure states. By utilizing the related symmetry, analytical results of GαααC are derived for any nnn-qubit GHZ states and W states, which show that the GHZ states are more genuinely entangled than the W states. With explicit examples, we demonstrate that the GαααC can distinguish different GME states that other GME measures fail to. These results justify the potential applications of GαααC in characterizing genuine multipartite entanglements.
Although genuine multipartite entanglement has already been generated and verified by experiments, most of the existing measures cannot detect genuine entanglement faithfully. In this work, by exploiting for the first time a previously overlooked constraint for the distribution of entanglement in three-qubit systems, we reveal a new genuine tripartite entanglement measure, which is related to the area of a so-called concurrence triangle. It is compared with other existing measures and is found superior to previous attempts for different reasons. A specific example is illustrated to show that two tripartite entanglement measures can be inequivalent due to the high dimensionality of the Hilbert space. The properties of the triangle measure make it a candidate in potential quantum tasks and available to be used in any multiparty entanglement problems.
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.