Monogamy Inequality in Terms of Entanglement Measures Based on Distance for Pure Multiqubit States
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Monogamy Inequality in Terms of Entanglement Measures Based on Distance for Pure Multiqubit States Limin Gao1 · Fengli Yan1
· Ting Gao2
Received: 22 May 2020 / Accepted: 31 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Using very general arguments, we prove that any entanglement measures based on distance must be maximal on pure states. Furthermore, we show that Bures measure of entanglement and geometric measure of entanglement satisfy the monogamy inequality on all pure multiqubit states. Finally, using the power of Bures measure of entanglement and geometric measure of entanglement, we present a class of tight monogamy relations for pure states of multiqubit systems. Keywords Monogamy inequality · Bures measure of entanglement · Geometric measure of entanglement · Multiqubit systems
Entanglement is one of the most important features present in quantum theory. An important property distinguishing entanglement from classical correlations 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. It means that entanglement cannot be freely shared unconditionally among the multipartite quantum systems. For instance, for a three partite quantum system A, B and C, if A and B share maximal entanglement, then they share no entanglement with C. MOE indicates that there is a trade-off on the amount of entanglement between the pairs AB and AC. MOE is very important in the context of quantum cryptography because it restricts on the amount of information that an eavesdropper could potentially obtain about the secret key extraction. As a matter of fact, many information-theoretic protocols [3–5] can be guaranteed secure by the constraints on the sharing of entanglement.
Fengli Yan
[email protected] Ting Gao [email protected] 1
College of Physics, Hebei Key Laboratory of Photophysics Research and Application , Hebei Normal University, Shijiazhuang, 050024, China
2
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China
International Journal of Theoretical Physics
In 2000 Coffman, Kundu, and Wootters proved the first mathematical characterization of MOE for three-qubit state in terms of squared concurrence, known as CKW-inequality [1]. Osborne and Verstraete generalized this inequality to arbitrary multiqubit systems [6]. Later, it was proved that the same monogamy inequalities hold for other entanglement measures [7–13]. These monogamy relations play a very important role in quantum information theory [14], condensed-matter physics [15] and even black-hole physics [16]. One class of entanglement measures based on distance was proposed in [17, 18]. Those measures quantify the minimum distance between a given state and the separable states. Examples of such measures are the Bures measure of entanglement [18] and geometric measure of entanglement [19], which are the widely used entanglement measures in
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