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      Lower and upper bounds for entanglement of Rényi- α entropy

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      Author(s): 1 , , 2 , 3 , , 1
      Publication date (Electronic):
      Journal: Scientific Reports
      Publisher: Nature Publishing Group UK

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          Abstract

          Entanglement Rényi- α entropy is an entanglement measure. It reduces to the standard entanglement of formation when α tends to 1. We derive analytical lower and upper bounds for the entanglement Rényi- α entropy of arbitrary dimensional bipartite quantum systems. We also demonstrate the application our bound for some concrete examples. Moreover, we establish the relation between entanglement Rényi- α entropy and some other entanglement measures.

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          Entanglement of Formation of an Arbitrary State of Two Qubits

          William Wootters (1997)
          The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state is defined as the minimum average entanglement of an ensemble of pure states that represents the given mixed state. An earlier paper [Phys. Rev. Lett. 78, 5022 (1997)] conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula to be true for a special class of mixed states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing pure-state decompositions.
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            Mixed State Entanglement and Quantum Error Correction

            William Wootters, John Smolin, David P. DiVincenzo … (1996)
            Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state \(|\xi\rangle\) can be transmitted at some rate Q through a noisy channel \(\chi\) without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state \(\hat{M}(\chi)\) (obtained by sharing halves of EPR pairs through a channel \(\chi\)) yields a QECC on \(\chi\) with rate \(Q=D\), and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts \(D_1(M)\) and \(D_2(M)\) that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for \(E(M)\) when \(M\) is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic \(Q\) (or \(D\)) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.
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              A computable measure of entanglement

              , (2001)
              We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary bipartite system. We show that it does not increase under local manipulations of the system, and use it to obtain a bound on the teleportation capacity and on the distillable entanglement of mixed states.
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                Author and article information

                Contributors
                Wei Song: wsong1@mail.ustc.edu.cn
                Lin Chen: linchen@buaa.edu.cn
                Journal
                Journal ID (nlm-ta): Sci Rep
                Journal ID (iso-abbrev): Sci Rep
                Title: Scientific Reports
                Publisher: Nature Publishing Group UK (London )
                ISSN (Electronic): 2045-2322
                Publication date (Electronic): 23 December 2016
                Publication date PMC-release: 23 December 2016
                Publication date Collection: 2016
                Volume: 6
                Electronic Location Identifier: 23
                Affiliations
                [1 ]ISNI 0000 0004 1761 5124, GRID grid.462326.7, Institute for Quantum Control and Quantum Information, and School of Electronic and Information Engineering, , Hefei Normal University, ; Hefei, 230601 China
                [2 ]ISNI 0000 0000 9999 1211, GRID grid.64939.31, School of Mathematics and Systems Science, , Beihang University, ; Beijing, 100191 China
                [3 ]ISNI 0000 0000 9999 1211, GRID grid.64939.31, International Research Institute for Multidisciplinary Science, , Beihang University, ; Beijing, 100191 China
                Article
                Publisher ID: 29
                DOI: 10.1038/s41598-016-0029-9
                PMC ID: 5431388
                PubMed ID: 28003661
                SO-VID: 194e4d08-0ade-4844-b6d2-0366142a7fa6
                Copyright © © The Author(s) 2016
                License:

                This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

                History
                Date received : 13 October 2016
                Date accepted : 11 November 2016
                Categories
                Subject: Article
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                issue-copyright-statement © The Author(s) 2016

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