Tighter monogamy and polygamy relations of multiparty quantum entanglement
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Tighter monogamy and polygamy relations of multiparty quantum entanglement Limin Gao1 · Fengli Yan1
· Ting Gao2
Received: 17 May 2020 / Accepted: 23 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We explore monogamy and polygamy relations of entanglement in multipartite systems. By using the power of the bipartite entanglement measure, we establish a class of tight monogamy relations of multiparty entanglement with larger lower bounds in comparison to all known entanglement monogamy relations. We also give a class of tight polygamy relations of multiparty entanglement with smaller upper bounds than the existing ones, in terms of the power of the entanglement of assistance. We provide examples in which our new monogamy and polygamy relations are tighter than the previous ones. Keywords Monogamy relation · Polygamy relation · Multiparty system · Quantum entanglement
1 Introduction One of the important properties of entanglement is the monogamy of entanglement (MOE) [1,2], which means that a quantum subsystem in a multipartite quantum system entangled with another subsystem limits its entanglement with the remaining ones. Thus, entanglement cannot be freely shared unconditionally among the multipartite quantum systems. For example, for a three-partite quantum system A, B and C, if A and B share maximal entanglement, then they share no entanglement with C. This indicates that it should obey some trade-off on the amount of entanglement between the pairs AB and AC.
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Fengli Yan [email protected] Ting Gao [email protected]
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College of Physics, Hebei Normal University, Shijiazhuang 050024, China
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School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China 0123456789().: V,-vol
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The monogamy of entanglement is fundamentally important in the context of quantum cryptography since it restricts on the amount of information that an eavesdropper could potentially obtain about the secret key extraction. In fact, many informationtheoretic protocols can be guaranteed secure by the constraints on the sharing of entanglement, such as quantum key distribution (QKD) protocols [3–5]. The first mathematical characterization of MOE was expressed as a form of inequality for three-qubit state in terms of squared concurrence [1], which was generalized to arbitrary multiqubit systems by Osborne and Verstraete [6]. Later, the same monogamy inequality was also generalized to other entanglement measures [7–13]. The monogamy inequality corresponds to a residual quantity. Using concurrence as a bipartite entanglement measure, the residual quantity corresponds to 3-tangle which has clear meanings. The residual quantity based on entanglement of formation is shown to act as a kind of indicator for multiqubit entanglement, which can detect all genuine multipartite entangled states [11]. These monogamy relations also play an important role in quantum information theory [14], condensed-matter physics [15] and even black-hole physics [
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