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Entanglement measures induced by fidelity-based distances

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Abstract

We propose entanglement measures by means of the fidelity-based distances. The fidelity-based distance here is analogous to the relative entropy to the well-known entanglement measure, entanglement of formation. Our approach can be defined for any multipartite systems in a unified way. We also show that these fidelity-based measures are entanglement monotones and the bipartite ones are monogamous.

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References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  2. Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009)

    MathSciNet  ADS  Google Scholar 

  3. Plenio, M.P., Virmani, S.: An introduction to entanglement measures. Quant. Inf. Comput. 7, 1–51 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53(4), 2046 (1996)

    ADS  Google Scholar 

  5. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)

    MathSciNet  MATH  ADS  Google Scholar 

  6. Horodecki, M.: Entanglement measures. Quant. Inf. Comput. 1, 3 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Donald, M.J., Horodecki, M., Rudolph, O.: The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252 (2002)

    MathSciNet  MATH  ADS  Google Scholar 

  8. Horodecki, M., Horodecki, P., Horodecki, R.: Limits for entanglement measures. Phys. Rev. Lett. 84, 2014 (2000)

    MATH  ADS  Google Scholar 

  9. Hayden, P.M., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A 34, 6891 (2001)

    MathSciNet  MATH  ADS  Google Scholar 

  10. Liang, Y., et al.: Quantum fidelity measures for mixed states. Rep. Prog. Phys. 82, 076001 (2019)

    MathSciNet  ADS  Google Scholar 

  11. Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–23 (1994)

    MathSciNet  MATH  ADS  Google Scholar 

  12. Uhlmann, A.: The ‘transition probability’ in the state space of a*-algebra. Rep. Math. Phys. 9, 273–9 (1976)

    MathSciNet  MATH  ADS  Google Scholar 

  13. Zhang, L., Chen, L., Bu, K.: Fidelity between one bipartite quantum state and another undergoing local unitary dynamics. Quant. Inf. Process. 14, 4715–4730 (2015)

    MathSciNet  MATH  ADS  Google Scholar 

  14. Fawzi, O., Renner, R.: Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys. 340(2), 575–611 (2015)

    MathSciNet  MATH  ADS  Google Scholar 

  15. Luo, S., Zhang, Q.: Informational distance on quantum state space. Phys. Rev. A 69, 032106 (2004)

    MathSciNet  ADS  Google Scholar 

  16. Ma, Z., Zhang, F.L., Chen, J.L.: Geometric interpretation for the a fidelity and its relation with the Bures fidelity. Phys. Rev. A 78, 064305 (2008)

    ADS  Google Scholar 

  17. Raggio, G.A.: Generalized Transition Probabilities and Applications Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, pp. 327–335. Springer, New York (1984)

    Google Scholar 

  18. Rastegin, A.E.: Sine distance for quantum states. arXiv:quant-ph/0602112

  19. Gilchrist, A., Langford, N.K., Nielsen, M.A.: Distance measures to compare real and ideal quantum processes. Phys. Rev. A 71, 062310 (2005)

    ADS  Google Scholar 

  20. Hübner, M.: Explicit computation of the Bures distance for density matrices. Phys. Lett. A 163, 239–42 (1992)

    MathSciNet  ADS  Google Scholar 

  21. Vedral, V., Plenio, M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57, 1619 (1998)

    ADS  Google Scholar 

  22. Shao, L.H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2015)

    ADS  Google Scholar 

  23. Xiong, C., Kumar, A., Wu, J.: Family of coherence measures and duality between quantum coherence and path distinguishability. Phys. Rev. A 98, 032324 (2018)

    ADS  Google Scholar 

  24. Xiong, C., Kumar, A., Huang, M., Das, S., Sen, U., Wu, J.: Partial coherence and quantum correlation with fidelity and affinity distances. Phys. Rev. A 99, 032305 (2019)

    ADS  Google Scholar 

  25. Liu, C.L., Zhang, D.J., Yu, X.D., Ding, Q.M., Liu, L.: A new coherence measure based on fidelity. Quant. Inf. Process. 16(8), 198 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  26. Terhal, B.: Is entanglement monogamous? IBM J. Res. Dev. 48, 71 (2004)

    Google Scholar 

  27. Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)

    ADS  Google Scholar 

  28. Dhar, H.S., Pal, A.K., Rakshit, D., De, A.S., Sen, U.: Monogamy of quantum correlations-a review. In: Lectures on General Quantum Correlations and Their Applications, pp. 23–64. Springer, Cham (2017)

  29. Guo, Y., Gour, G.: Monogamy of the entanglement of formation. Phys. Rev. A 99, 042305 (2019)

    ADS  Google Scholar 

  30. Gour, G., Guo, Y.: Monogamy of entanglement without inequalities. Quantum 2, 81 (2018)

    Google Scholar 

  31. Guo, Y.: Strict entanglement monotonicity under local operations and classical communication. Phys. Rev. A 99, 022338 (2019)

    ADS  Google Scholar 

  32. Koashi, M., Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)

    MathSciNet  ADS  Google Scholar 

  33. Guo, Y., Zhang, L.: Multipartite entanglement measure and complete monogamy relation. Phys. Rev. A 101, 032301 (2020)

    MathSciNet  ADS  Google Scholar 

  34. Vidal, G.: Entanglement monotones. J. Mod. Opt. 47, 355 (2000)

    MathSciNet  ADS  Google Scholar 

  35. Zhang, L., Fei, S.: Quantum fidelity and relative entropy between unitary orbits. J. Phys. A: Math. Theor. 47, 055301 (2014)

    MathSciNet  MATH  ADS  Google Scholar 

  36. Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    ADS  Google Scholar 

  37. Szalay, S.: Multipartite entanglement measures. Phys. Rev. A 92, 042329 (2015)

    ADS  Google Scholar 

  38. Kim, I.K.: Modulus of convexity for operator convex functions. J. Math. Phys. 55, 082201 (2014)

    MathSciNet  MATH  ADS  Google Scholar 

  39. Nielsen, M.A., Chuang, I.L.: Quantum Computatation and Quantum Information. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  40. Hong, Y., Gao, T., Yan, F.: Measure of multipartite entanglement with computable lower bounds. Phys. Rev. A 86, 062323 (2012)

    ADS  Google Scholar 

  41. Hiesmayr, B.C., Huber, M.: Multipartite entanglement measure for all discrete systems. Phys. Rev. A 78, 012342 (2008)

    ADS  Google Scholar 

  42. Horodecki, M.: Simplifying monotonicity conditions for entanglement measures. Open Syst. Inform. Dyn. 12(03), 231–237 (2005)

    MathSciNet  MATH  Google Scholar 

  43. Yang, D., Horodecki, K., Horodecki, M., et al.: Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof. IEEE Trans. Inf. Theory 55(7), 3375–3387 (2009)

    MathSciNet  MATH  Google Scholar 

  44. Yang, D., Horodecki, M., Wang, Z.D.: An additive and operational entanglement measure: conditional entanglement of mutual information. Phys. Rev. Lett. 101, 140501 (2008)

    ADS  Google Scholar 

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Acknowledgements

Y.G. was supported by the National Natural Science Foundation of China under Grant No. 11971277, the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi, and the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province under Grant No. 2019KJ034. L.Z. is supported by the National Natural Science Foundation of China under Grant No. 11971140, and also by the Zhejiang Provincial Natural Science Foundation of China under grant No. LY17A010027 and the National Natural Science Foundation of China under Grant Nos. 11701259 and 61771174.

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Authors and Affiliations

  1. Institute of Quantum Information Science, Shanxi Datong University, Datong, 037009, Shanxi, China

    Yu Guo

  2. School of Mathematics and Statistics, Shanxi Datong University, Datong, 037009, China

    Yu Guo & Huting Yuan

  3. Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China

    Lin Zhang

  4. Max-Planck-Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Lin Zhang

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  1. Yu Guo
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  2. Lin Zhang
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  3. Huting Yuan
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Correspondence to Yu Guo or Lin Zhang.

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Guo, Y., Zhang, L. & Yuan, H. Entanglement measures induced by fidelity-based distances. Quantum Inf Process 19, 282 (2020). https://doi.org/10.1007/s11128-020-02787-2

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