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Abstract
We study the ground-state entanglement in the quantum Ising model with nearest neighbor ferromagnetic coupling J and find a sequential increase of entanglement depth d with growing J. This entanglement avalanche starts with two-point entanglement, as measured by the concurrence, and continues via the three-tangle and four-tangle, until finally, deep in the ferromagnetic phase for J = ∞, arriving at a pure L-partite (GHZ type) entanglement of all L spins. Comparison with the two, three, and four-point correlations reveals a similar sequence and shows strong ties to the above entanglement measures for small J. However, we also find a partial inversion of the hierarchy, where the four-point correlation exceeds the three- and two-point correlations, well before the critical point is reached. Qualitatively similar behavior is also found for the Bose-Hubbard model, suggesting that this is a general feature of a quantum phase transition. This should be taken into account in the approximations starting from a mean-field limit.
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Affiliation(s)
- Konstantin V Krutitsky
- Institut für Theoretische Physik, Universität Duisburg-Essen, D-47048, Duisburg, Germany.
| | - Andreas Osterloh
- Institut für Theoretische Physik, Universität Duisburg-Essen, D-47048, Duisburg, Germany
| | - Ralf Schützhold
- Institut für Theoretische Physik, Universität Duisburg-Essen, D-47048, Duisburg, Germany
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Entanglement distribution in multi-particle systems in terms of unified entropy. Sci Rep 2017; 7:1122. [PMID: 28442724 PMCID: PMC5430764 DOI: 10.1038/s41598-017-01286-2] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2017] [Accepted: 03/28/2017] [Indexed: 11/08/2022] Open
Abstract
We investigate the entanglement distribution in multi-particle systems in terms of unified (q, s)-entropy. We find that for any tripartite mixed state, the unified (q, s)-entropy entanglement of assistance follows a polygamy relation. This polygamy relation also holds in multi-particle systems. Furthermore, a generalized monogamy relation is provided for unified (q, s)-entropy entanglement in the multi-qubit system.
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Lami L, Hirche C, Adesso G, Winter A. Schur Complement Inequalities for Covariance Matrices and Monogamy of Quantum Correlations. PHYSICAL REVIEW LETTERS 2016; 117:220502. [PMID: 27925721 DOI: 10.1103/physrevlett.117.220502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2016] [Indexed: 06/06/2023]
Abstract
We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. This allows us to prove general results on the monogamy of entanglement and steering quantifiers in continuous variable systems with an arbitrary number of modes per party. A powerful hierarchical relation for correlation measures based on the log-determinant of covariance matrices is further established for all Gaussian states, which has no counterpart among quantities based on the conventional von Neumann entropy.
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Affiliation(s)
- Ludovico Lami
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
| | - Christoph Hirche
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
| | - Gerardo Adesso
- Centre for the Mathematical and Theoretical Physics of Quantum Non-Equilibrium Systems, School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Andreas Winter
- Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
- ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluis Companys 23, ES-08010 Barcelona, Spain
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Generalised monogamy relation of convex-roof extended negativity in multi-level systems. Sci Rep 2016; 6:36700. [PMID: 27857163 PMCID: PMC5114565 DOI: 10.1038/srep36700] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2016] [Accepted: 10/19/2016] [Indexed: 12/02/2022] Open
Abstract
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.
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Lancien C, Di Martino S, Huber M, Piani M, Adesso G, Winter A. Should Entanglement Measures be Monogamous or Faithful? PHYSICAL REVIEW LETTERS 2016; 117:060501. [PMID: 27541447 DOI: 10.1103/physrevlett.117.060501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2016] [Indexed: 06/06/2023]
Abstract
"Is entanglement monogamous?" asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum information theory. While the answer is affirmative in the qualitative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.
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Affiliation(s)
- Cécilia Lancien
- Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
- Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
| | - Sara Di Martino
- Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
| | - Marcus Huber
- Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
- Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
- Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
| | - Marco Piani
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Gerardo Adesso
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Andreas Winter
- Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
- ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluis Companys 23, ES-08010 Barcelona, Spain
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Yuan GM, Song W, Yang M, Li DC, Zhao JL, Cao ZL. Monogamy relation of multi-qubit systems for squared Tsallis-q entanglement. Sci Rep 2016; 6:28719. [PMID: 27346605 PMCID: PMC4921912 DOI: 10.1038/srep28719] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2016] [Accepted: 06/06/2016] [Indexed: 11/09/2022] Open
Abstract
Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.
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Affiliation(s)
- Guang-Ming Yuan
- School of Physics and Material Science, Anhui University, Hefei, 230601, China
| | - Wei Song
- Institute for Quantum Control and Quantum Information; School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601, China
| | - Ming Yang
- School of Physics and Material Science, Anhui University, Hefei, 230601, China
| | - Da-Chuang Li
- Institute for Quantum Control and Quantum Information; School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601, China
| | - Jun-Long Zhao
- School of Physics and Material Science, Anhui University, Hefei, 230601, China
| | - Zhuo-Liang Cao
- Institute for Quantum Control and Quantum Information; School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601, China
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Regula B, Adesso G. Entanglement Quantification Made Easy: Polynomial Measures Invariant under Convex Decomposition. PHYSICAL REVIEW LETTERS 2016; 116:070504. [PMID: 26943522 DOI: 10.1103/physrevlett.116.070504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2015] [Indexed: 06/05/2023]
Abstract
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
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Affiliation(s)
- Bartosz Regula
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Gerardo Adesso
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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Eltschka C, Siewert J. Monogamy equalities for qubit entanglement from Lorentz invariance. PHYSICAL REVIEW LETTERS 2015; 114:140402. [PMID: 25910095 DOI: 10.1103/physrevlett.114.140402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2014] [Indexed: 06/04/2023]
Abstract
A striking result from nonrelativistic quantum mechanics is the monogamy of entanglement, which states that a particle can be maximally entangled only with one other party, not with several ones. While there is the exact quantitative relation for three qubits and also several inequalities describing monogamy properties, it is not clear to what extent exact monogamy relations are a general feature of quantum mechanics. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. They come about through the curious relationship between the nonrelativistic quantum mechanics of qubits and Minkowski space. We elucidate the origin of entanglement monogamy from this symmetry perspective and provide recipes to construct new families of such equalities.
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Affiliation(s)
- Christopher Eltschka
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Jens Siewert
- Departamento de Química Física, Universidad del País Vasco-Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
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