Monogamy of entanglement without inequalities

Gilad Gour1 and Yu Guo2

1Department of Mathematics and Statistics and Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
2Institute of Quantum Information Science, Shanxi Datong University, Datong, Shanxi 037009, China

We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.

► BibTeX data

► References

[1] V. Coffman, J. Kundu, and W. K. Wootters. Distributed entanglement. Phys. Rev. A, 61:052306, 2000. doi:10.1103/​PhysRevA.61.052306.
https://doi.org/10.1103/PhysRevA.61.052306

[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement. Rev. Mod. Phys., 81:865, 2009. doi:10.1103/​RevModPhys.81.865.
https://doi.org/10.1103/RevModPhys.81.865

[3] M. Koashi and A. Winter. Monogamy of quantum entanglement and other correlations. Phys. Rev. A, 69:022309, 2004. doi:10.1103/​PhysRevA.69.022309.
https://doi.org/10.1103/PhysRevA.69.022309

[4] G. Gour, D. A. Meyer, and B. C. Sanders. Deterministic entanglement of assistance and monogamy constraints. Phys. Rev. A, 72:042329, 2005. doi:10.1103/​PhysRevA.72.042329.
https://doi.org/10.1103/PhysRevA.72.042329

[5] T. J. Osborne and F. Verstraete. General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett., 96:220503, 2006. doi:10.1103/​PhysRevLett.96.220503.
https://doi.org/10.1103/PhysRevLett.96.220503

[6] Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativity for three-qubit states. Phys. Rev. A, 75:062308, 2007. doi:10.1103/​PhysRevA.75.062308.
https://doi.org/10.1103/PhysRevA.75.062308

[7] J. S. Kim, A. Das, and B. C. Sanders. Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity. Phys. Rev. A, 79:012329, 2009. doi:10.1103/​PhysRevA.79.012329.
https://doi.org/10.1103/PhysRevA.79.012329

[8] X. N. Zhu and S. M. Fei. Entanglement monogamy relations of qubit systems. Phys. Rev. A, 90: 024304, 2014. doi:10.1103/​PhysRevA.90.024304.
https://doi.org/10.1103/PhysRevA.90.024304

[9] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang. General monogamy relation for the entanglement of formation in multiqubit systems. Phys. Rev. Lett., 113:100503, 2014. doi:10.1103/​PhysRevLett.113.100503.
https://doi.org/10.1103/PhysRevLett.113.100503

[10] J. H. Choi and J. S. Kim. Negativity and strong monogamy of multiparty quantum entanglement beyond qubits. Phys. Rev. A, 92:042307, 2015. doi:10.1103/​PhysRevA.92.042307.
https://doi.org/10.1103/PhysRevA.92.042307

[11] Y. Luo and Y. Li. Monogamy of $\alpha$th power entanglement measurement in qubit systems. Ann. Phys., 362:511-520, 2015. doi:10.1016/​j.aop.2015.08.022.
https://doi.org/10.1016/j.aop.2015.08.022

[12] X. N. Zhu and S. M. Fei. Entanglement monogamy relations of concurrence for $N$-qubit systems. Phys. Rev. A, 92:062345, 2015. doi:10.1103/​PhysRevA.92.062345.
https://doi.org/10.1103/PhysRevA.92.062345

[13] A. Kumar. Conditions for monogamy of quantum correlations in multipartite systems. Phys. Lett. A, 380:3044-3050, 2016. doi:10.1016/​j.physleta.2016.07.032.
https://doi.org/10.1016/j.physleta.2016.07.032

[14] G. Gour, S. Bandyopadhyay, and B. C. Sanders. Dual monogamy inequality for entanglement. J. Math. Phys., 48:012108, 2007. doi:10.1063/​1.2435088.
https://doi.org/10.1063/1.2435088

[15] Y.-C. Ou. Violation of monogamy inequality for higher dimensional objects. Phys. Rev. A, 75:034305, 2007. doi:10.1103/​PhysRevA.75.034305.
https://doi.org/10.1103/PhysRevA.75.034305

[16] T. Hiroshima, G. Adesso and F. Illuminati. Monogamy inequality for distributed Gaussian entanglement. Phys. Rev. Lett., 98:050503, 2007. doi:10.1103/​PhysRevLett.98.050503.
https://doi.org/10.1103/PhysRevLett.98.050503

[17] G. Adesso and F. Illuminati. Strong monogamy of bipartite and genuine multipartite entanglement: The Gaussian case. Phys. Rev. Lett., 99:150501, 2007. doi:10.1103/​PhysRevLett.99.150501.
https://doi.org/10.1103/PhysRevLett.99.150501

[18] J. S. Kim and B. C. Sanders. Generalized W-class state and its monogamy relation. J. Phys. A, 41:495301, 2008. doi:10.1088/​1751-8113/​41/​49/​495301.
https://doi.org/10.1088/1751-8113/41/49/495301

[19] J. S. Kim and B. C. Sanders. Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys. A, 43:445305, 2010. doi:10.1088/​1751-8113/​43/​44/​445305.
https://doi.org/10.1088/1751-8113/43/44/445305

[20] X.-J. Ren and W. Jiang. Entanglement monogamy inequality in a $2\otimes 2\otimes 4$ system. Phys. Rev. A, 81:024305, 2010. doi:10.1103/​PhysRevA.81.024305.
https://doi.org/10.1103/PhysRevA.81.024305

[21] M. F. Cornelio and M. C. de Oliveira. Strong superadditivity and monogamy of the Rényi measure of entanglement. Phys. Rev. A, 81:032332, 2010. doi:10.1103/​PhysRevA.81.032332.
https://doi.org/10.1103/PhysRevA.81.032332

[22] A. Streltsov, G. Adesso, M. Piani, and D. Bruß. Are general quantum correlations monogamous? Phys. Rev. Lett., 109:050503, 2012. doi:10.1103/​PhysRevLett.109.050503.
https://doi.org/10.1103/PhysRevLett.109.050503

[23] M. F. Cornelio. Multipartite monogamy of the concurrence. Phys. Rev. A, 87:032330, 2013. doi:10.1103/​PhysRevA.87.032330.
https://doi.org/10.1103/PhysRevA.87.032330

[24] S.-Y. Liu, B. Li, W.-L. Yang, and H. Fan. Monogamy deficit for quantum correlations in a multipartite quantum system. Phys. Rev. A, 87:062120, 2013. doi:10.1103/​PhysRevA.87.062120.
https://doi.org/10.1103/PhysRevA.87.062120

[25] T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini. Monogamy of entanglement of formation. Phys. Rev. A, 89:034303, 2014. doi:10.1103/​PhysRevA.89.034303.
https://doi.org/10.1103/PhysRevA.89.034303

[26] B. Regula, S. D. Martino, S. Lee, and G. Adesso. Strong monogamy conjecture for multiqubit entanglement: the four-qubit case. Phys. Rev. Lett., 113:110501, 2014. doi:10.1103/​PhysRevLett.113.110501.
https://doi.org/10.1103/PhysRevLett.113.110501

[27] K. Salini, R. Prabhub, Aditi Sen(De), and Ujjwal Sen. Monotonically increasing functions of any quantum correlation can make all multiparty states monogamous. Ann. Phys., 348:297-305, 2014. doi:10.1016/​j.aop.2014.06.001.
https://doi.org/10.1016/j.aop.2014.06.001

[28] H. He and G. Vidal. Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A, 91:012339, 2015. doi:10.1103/​PhysRevA.91.012339.
https://doi.org/10.1103/PhysRevA.91.012339

[29] C. Eltschka and J. Siewert. Monogamy equalities for qubit entanglement from Lorentz invariance. Phys. Rev. Lett., 114:140402, 2015. doi:10.1103/​PhysRevLett.114.140402.
https://doi.org/10.1103/PhysRevLett.114.140402

[30] A. Kumar, R. Prabhu, A. Sen(de), and U. Sen. Effect of a large number of parties on the monogamy of quantum correlations. Phys. Rev. A, 91:012341, 2015. doi:10.1103/​PhysRevA.91.012341.
https://doi.org/10.1103/PhysRevA.91.012341

[31] Lancien et al. Should entanglement measures be monogamous or faithful? Phys. Rev. Lett., 117:060501, 2016. doi:10.1103/​PhysRevLett.117.060501.
https://doi.org/10.1103/PhysRevLett.117.060501

[32] L. Lami, C. Hirche, G. Adesso, and A. Winter. Schur complement inequalities for covariance matrices and monogamy of quantum correlations. Phys. Rev. Lett., 117:220502, 2016. doi:10.1103/​PhysRevLett.117.220502.
https://doi.org/10.1103/PhysRevLett.117.220502

[33] Song et al. General monogamy relation of multiqubit systems in terms of squared Rényi-$\alpha$ entanglement. Phys. Rev. A, 93:022306, 2016. doi:10.1103/​PhysRevE.93.022306.
https://doi.org/10.1103/PhysRevE.93.022306

[34] B. Regula, A. Osterloh, and G. Adesso. Strong monogamy inequalities for four qubits. Phys. Rev. A, 93:052338, 2016. doi:10.1103/​PhysRevA.93.052338.
https://doi.org/10.1103/PhysRevA.93.052338

[35] Y. Luo, T. Tian, L.-H. Shao, and Y. Li. General monogamy of Tsallis $q$-entropy entanglement in multiqubit systems. Phys. Rev. A, 93: 062340, 2016. doi:10.1103/​PhysRevA.93.062340.
https://doi.org/10.1103/PhysRevA.93.062340

[36] E. Jung and D. Park. Testing the monogamy relations via rank-2 mixtures. Phys. Rev. A, 94:042330, 2016. doi:10.1103/​PhysRevA.94.042330.
https://doi.org/10.1103/PhysRevA.94.042330

[37] S. Cheng and M. J. W. Hall. Anisotropic invariance and the distribution of quantum correlations. Phys. Rev. Lett., 118:010401, 2017. doi:10.1103/​PhysRevLett.118.010401.
https://doi.org/10.1103/PhysRevLett.118.010401

[38] G. W. Allen and D. A. Meyer. Polynomial monogamy relations for entanglement negativity. Phys. Rev. Lett., 118: 080402, 2017. doi:10.1103/​PhysRevLett.118.080402.
https://doi.org/10.1103/PhysRevLett.118.080402

[39] Q. Li, J. Cui, S. Wang, and G.-L. Long. Entanglement monogamy in three qutrit systems. Sci. Rep., 7:1946, 2017. doi:10.1038/​s41598-017-02066-8.
https://doi.org/10.1038/s41598-017-02066-8

[40] S. Camalet. Monogamy Inequality for any local quantum resource and entanglement. Phys. Rev. Lett., 119: 110503, 2017. doi:10.1103/​PhysRevLett.119.110503.
https://doi.org/10.1103/PhysRevLett.119.110503

[41] B. M. Terhal. IBM Journal of Research and Development,48(1):71-78, 2004. doi:10.1147/​rd.481.0071.
https://doi.org/10.1147/rd.481.0071

[42] M. Pawlowski. Security proof for cryptographic protocols based only on the monogamy of Bell’s inequality violations. Phys. Rev. A, 82:032313, 2010. doi:10.1103/​PhysRevA.82.032313.
https://doi.org/10.1103/PhysRevA.82.032313

[43] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden. Quantum cryptography. Rev. Mod. Phys., 74:145, 2002. doi:10.1103/​RevModPhys.74.145.
https://doi.org/10.1103/RevModPhys.74.145

[44] W. Dür, G. Vidal, and J. I. Cirac. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A, 62:062314, 2000. doi:10.1103/​PhysRevA.62.062314. doi:10.1103/​PhysRevA.62.062314.
https://doi.org/10.1103/PhysRevA.62.062314

[45] G. L. Giorgi. Monogamy properties of quantum and classical correlations. Phys. Rev. A, 84: 054301, 2011. doi:10.1103/​PhysRevA.84.054301.
https://doi.org/10.1103/PhysRevA.84.054301

[46] R. Prabhu, A. K. Pati, A. Sen(De), and U. Sen. Conditions for monogamy of quantum correlations: Greenberger-Horne-Zeilinger versus W states. Phys. Rev. A, 85:040102(R), 2012. doi:10.1103/​PhysRevA.85.040102.
https://doi.org/10.1103/PhysRevA.85.040102

[47] Ma et al. Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems. Nat. Phys., 7:399, 2011. doi:10.1038/​nphys1919.
https://doi.org/10.1038/nphys1919

[48] F. G. S. L. Brandao and A. W. Harrow, in Proceedings of the 45th Annual ACM Symposium on Theory of Computing, 2013. http:/​/​dl.acm.org/​citation.cfm?doid=2488608.
http:/​/​dl.acm.org/​citation.cfm?doid=2488608

[49] A. García-Sáez and J. I. Latorre. Renormalization group contraction of tensor networks in three dimensions. Phys. Rev. B, 87:085130, 2013. doi:10.1103/​PhysRevB.87.085130.
https://doi.org/10.1103/PhysRevB.87.085130

[50] Rao et al. Multipartite quantum correlations reveal frustration in a quantum Ising spin system. Phys. Rev. A, 88:022312, 2013. doi:10.1103/​PhysRevA.88.022312.
https://doi.org/10.1103/PhysRevA.88.022312

[51] C. H. Bennett, in Proceedings of the FQXi 4th International Conference, Vieques Island, Puerto Rico, 2014, http:/​/​fqxi.org/​conference/​talks/​2014.
http:/​/​fqxi.org/​conference/​talks/​2014

[52] L. Susskind. Black hole complementarity and the Harlow-Hayden conjecture. https:/​/​arxiv.org/​abs/​1301.4505.
arXiv:1301.4505

[53] S. Lloyd and J. Preskill. Unitarity of black hole evaporation in final-state projection models. J. High Energy Phys., 08:126, 2014. doi:10.1007/​JHEP08(2014)126.
https://doi.org/10.1007/JHEP08(2014)126

[54] P. W. Shor, J. A. Smolin, and B. M. Terhal. Nonadditivity of bipartite distillable entanglement follows from a conjecture on bound entangled Werner states. Phys. Rev. Lett., 86:2681–2684, 2001. doi:10.1103/​PhysRevLett.86.2681.
https://doi.org/10.1103/PhysRevLett.86.2681

[55] P. W. Shor. Equivalence of additivity questions in quantum information theory. Commun. Math. Phys., 246(3):453-472, 2004. doi:10.1007/​s00220-003-0981-7.
https://doi.org/10.1007/s00220-003-0981-7

[56] K. G. H. Vollbrecht and R. F. Werner. Entanglement measures under symmetry. Phys. Rev. A, 64:062307, 2001. doi:10.1103/​PhysRevA.64.062307.
https://doi.org/10.1103/PhysRevA.64.062307

[57] G. Gour. Family of concurrence monotones and its applications. Phys. Rev. A, 71:012318, 2005. doi:10.1103/​PhysRevA.71.012318.
https://doi.org/10.1103/PhysRevA.71.012318

[58] DiVincenzo et al. Entanglement of Assistance. Lecture Notes in Computer Science, 1509:247, 1999. doi:10.1007/​3-540-49208-9_21.
https://doi.org/10.1007/3-540-49208-9_21

[59] P. Hayden, R. Jozsa, D. Petz, and A. Winter. Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys., 246(2):359-374, 2004. doi:10.1007/​s00220-004-1049-z.
https://doi.org/10.1007/s00220-004-1049-z

[60] M. B. Plenio. Logarithmic negativity: a full entanglement monotone that is not convex. Phys. Rev. Lett., 95:090503, 2005. doi:10.1103/​PhysRevLett.95.090503. Erratum Phys. Rev. Lett., 95:119902, 2005. doi:10.1103/​PhysRevLett.95.119902.
https://doi.org/10.1103/PhysRevLett.95.090503.

[61] G. Vidal. Entanglement monotone. J. Mod. Opt., 47:355, 2000. doi:10.1080/​09500340008244048.
https://doi.org/10.1080/09500340008244048

[62] G. Gour and R. W. Spekkens. Entanglement of assistance is not a bipartite measure nor a tripartite monotone. Phys. Rev. A, 73:062331, 2006. doi:10.1103/​PhysRevA.73.062331.
https://doi.org/10.1103/PhysRevA.73.062331

[63] M. Gerstenhaber. On nilalgebras and linear varieties of nilpotent matrices (I). Amer. J. Math., 80:614-622, 1958. doi:10.2307/​2372773.
https://doi.org/10.2307/2372773

[64] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 80:2245, 1998. doi:10.1103/​PhysRevLett.80.2245.
https://doi.org/10.1103/PhysRevLett.80.2245

[1] H. He and G. Vidal. Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A, 91:012339, 2015. doi:10.1103/​PhysRevA.91.012339.
https://doi.org/10.1103/PhysRevA.91.012339

[2] C. Eltschka and J. Siewert. Monogamy equalities for qubit entanglement from Lorentz invariance. Phys. Rev. Lett., 114:140402, 2015. doi:10.1103/​PhysRevLett.114.140402.
https://doi.org/10.1103/PhysRevLett.114.140402

[3] K. M. R. Audenaert. On a block matrix inequality quantifying the monogamy of the negativity of entanglement. Lin. Multilin. Alg., 63(12):2526-2536, 2015. doi/​full/​10.1080/​03081087.2015.1024193.
https://doi.org/10.1080/03081087.2015.1024193

[4] Lancien et al. Should entanglement measures be monogamous or faithful? Phys. Rev. Lett., 117:060501, 2016. doi:10.1103/​PhysRevLett.117.060501.
https://doi.org/10.1103/PhysRevLett.117.060501

[5] S. Cheng and M. J. W. Hall. Anisotropic invariance and the distribution of quantum correlations. Phys. Rev. Lett., 118:010401, 2017. doi:10.1103/​PhysRevLett.118.010401.
https://doi.org/10.1103/PhysRevLett.118.010401

[6] G. W. Allen and D. A. Meyer. Polynomial monogamy relations for entanglement negativity. Phys. Rev. Lett., 118: 080402, 2017. doi:10.1103/​PhysRevLett.118.080402.
https://doi.org/10.1103/PhysRevLett.118.080402

[7] S. Hill and W. K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78:5022, 1997. doi:10.1103/​PhysRevLett.78.5022.
https://doi.org/10.1103/PhysRevLett.78.5022

[8] M. Koashi and A. Winter. Monogamy of quantum entanglement and other correlations. Phys. Rev. A, 69:022309, 2004. doi:10.1103/​PhysRevA.69.022309.
https://doi.org/10.1103/PhysRevA.69.022309

[9] V. Coffman, J. Kundu, and W. K. Wootters. Distributed entanglement. Phys. Rev. A, 61:052306, 2000. doi:10.1103/​PhysRevA.61.052306.
https://doi.org/10.1103/PhysRevA.61.052306

[10] X. N. Zhu and S. M. Fei, Phys. Rev. A 90, 024304 (2014).

[11] T. J. Osborne and F. Verstraete. General mnogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett., 96:220503, 2006. doi:10.1103/​PhysRevLett.96.220503.
https://doi.org/10.1103/PhysRevLett.96.220503

[12] Y.-C. Ou. Violation of monogamy inequality for higher dimensional objects. Phys. Rev. A, 75:034305, 2007. doi:10.1103/​PhysRevA.75.034305.
https://doi.org/10.1103/PhysRevA.75.034305

[13] X.-J. Ren and W. Jiang. Entanglement monogamy inequality in a $2\otimes 2\otimes 4$ system. Phys. Rev. A, 81:024305, 2010. doi:10.1103/​PhysRevA.81.024305.
https://doi.org/10.1103/PhysRevA.81.024305

[14] J. S. Kim and B. C. Sanders. Generalized W-class state and its monogamy relation. J. Phys. A, 41:495301, 2008. doi:10.1088/​1751-8113/​41/​49/​495301.
https://doi.org/10.1088/1751-8113/41/49/495301

[15] Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativity for three-qubit states. Phys. Rev. A, 75:062308, 2007. doi:10.1103/​PhysRevA.75.062308.
https://doi.org/10.1103/PhysRevA.75.062308

[16] Y. Luo and Y. Li. Monogamy of $\alpha$th power entanglement measurement in qubit systems. Ann. Phys., 362:511-520, 2015. doi:10.1016/​j.aop.2015.08.022.
https://doi.org/10.1016/j.aop.2015.08.022

[17] J. S. Kim, A. Das, and B. C. Sanders. Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity. Phys. Rev. A, 79:012329, 2009. doi:10.1103/​PhysRevA.79.012329.
https://doi.org/10.1103/PhysRevA.79.012329

[18] J. H. Choi and J. S. Kim. Negativity and strong monogamy of multiparty quantum entanglement beyond qubits. Phys. Rev. A, 92:042307, 2015. doi:10.1103/​PhysRevA.92.042307.
https://doi.org/10.1103/PhysRevA.92.042307

[19] A. Kumar. Conditions for monogamy of quantum correlations in multipartite systems. Phys. Lett. A, 380:3044-3050, 2016. doi:10.1016/​j.physleta.2016.07.032.
https://doi.org/10.1016/j.physleta.2016.07.032

[20] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang. General monogamy relation for the entanglement of formation in multiqubit systems. Phys. Rev. Lett., 113:100503, 2014. doi:10.1103/​PhysRevLett.113.100503.
https://doi.org/10.1103/PhysRevLett.113.100503

[21] T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini. Monogamy of entanglement of formation. Phys. Rev. A, 89:034303, 2014. doi:10.1103/​PhysRevA.89.034303.
https://doi.org/10.1103/PhysRevA.89.034303

[22] J. S. Kim. Tsallis entropy and general polygamy of multiparty quantum entanglement in arbitrary dimensions. Phys. Rev. A, 94:062338, 2016. doi:10.1103/​PhysRevA.94.062338.
https://doi.org/10.1103/PhysRevA.94.062338

[23] Y. Luo, T. Tian, L.-H. Shao, and Y. Li. General monogamy of Tsallis $q$-entropy entanglement in multiqubit systems. Phys. Rev. A, 93: 062340, 2016. doi:10.1103/​PhysRevA.93.062340.
https://doi.org/10.1103/PhysRevA.93.062340

[24] J. S. Kim and B. C. Sanders. Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys. A, 43:445305, 2010. doi:10.1088/​1751-8113/​43/​44/​445305.
https://doi.org/10.1088/1751-8113/43/44/445305

[25] M. F. Cornelio and M. C. de Oliveira. Strong superadditivity and monogamy of the Rényi measure of entanglement. Phys. Rev. A, 81:032332, 2010. doi:10.1103/​PhysRevA.81.032332.
https://doi.org/10.1103/PhysRevA.81.032332

[26] Song et al. General monogamy relation of multiqubit systems in terms of squared Rényi-$\alpha$ entanglement. Phys. Rev. A, 93:022306, 2016. doi:10.1103/​PhysRevE.93.022306.
https://doi.org/10.1103/PhysRevE.93.022306

[27] Y. Guo, J. Hou, and Y. Wang. Concurrence for infinite-dimensional quantum systems. Quant. Inf. Process., 12:2641-2653, 2013. doi:10.1007/​s11128-013-0552-6.
https://doi.org/10.1007/s11128-013-0552-6

[28] Y. Guo and J. Hou. Entanglement detection beyond the CCNR criterion for infinite-dimensions. Chin. Sci. Bull., 58(11):1250-1255, 2013. doi:10.1007/​s11434-013-5738-x.
https://doi.org/10.1007/s11434-013-5738-x

[29] M. J. Donald and M. Horodecki. Continuity of relative entropy of entanglement. Phys. Lett. A, 264:257, 1999. doi:10.1016/​S0375-9601(99)00813-0.
https://doi.org/10.1016/S0375-9601(99)00813-0

[30] The continuty of the convex roof extended entanglement measure can be checked according to Proposition 2 in supGuo2013qip, the continuty of partial trace and partial transpose is proved in supGuo2013csb, the continuty of the realtive entropy entanglement is proved in supDonald1999pla.

[31] P. Hayden, R. Jozsa, D. Petz, and A. Winter. Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys., 246(2):359-374, 2004. doi:10.1007/​s00220-004-1049-z.
https://doi.org/10.1007/s00220-004-1049-z

[32] G. Gour and R. W. Spekkens. Entanglement of assistance is not a bipartite measure nor a tripartite monotone. Phys. Rev. A, 73:062331, 2006. doi:10.1103/​PhysRevA.73.062331.
https://doi.org/10.1103/PhysRevA.73.062331

[33] A. Uhlmann. Roofs and Convexity. Entropy, 12:1799-1832, 2010. doi:10.3390/​e12071799.
https://doi.org/10.3390/e12071799

[34] M. Gerstenhaber. On nilalgebras and linear varieties of nilpotent matrices (I). Amer. J. Math., 80:614-622, 1958. doi:10.2307/​2372773.
https://doi.org/10.2307/2372773

[35] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 80:2245, 1998. doi:10.1103/​PhysRevLett.80.2245.
https://doi.org/10.1103/PhysRevLett.80.2245

[36] G. Gour, D. A. Meyer, and B. C. Sanders. Deterministic entanglement of assistance and monogamy constraints. Phys. Rev. A, 72:042329, 2005. doi:10.1103/​PhysRevA.72.042329.
https://doi.org/10.1103/PhysRevA.72.042329

[37] T. Laustsen, F. Verstraete, and S. J. van Enk. Local vs. joint measurements for the entanglement of assistance. Quant. Inf. Comput., 3:64, 2003. arXiv:0206192.
arXiv:quant-ph/0206192

► Cited by (beta)

Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.