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Triangle-like inequalities related to coherence and entanglement negativity

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Abstract

Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the \(l_1\)-norm measure of coherence for convex combination of arbitrary n pure states of a quantum state \(\rho \). Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in Dai et al. (Phys. Rev. A 96:062308, 2017). Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities.

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References

  1. Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)

    ADS  Google Scholar 

  2. Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)

    MathSciNet  ADS  Google Scholar 

  3. Girolami, D.: Observable measure of quantum coherence in finite dimensional systems. Phys. Rev. Lett. 113, 170401 (2014)

    Article  ADS  Google Scholar 

  4. Yao, Y., Xiao, X., Ge, L., Sun, C.P.: Quantum coherence in multipartite systems. Phys. Rev. A 92, 022112 (2015)

    Article  ADS  Google Scholar 

  5. Yu, C.S., Song, H.S.: Bipartite concurrence and localized coherence. Phys. Rev. A 80, 022324 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  6. Rastegin, A.E.: Quantum-coherence quantifiers based on the Tsallis relative \(\alpha \) entropies. Phys. Rev. A 93, 032136 (2016)

    Article  ADS  Google Scholar 

  7. Ma, J., Yadin, B., Girolami, D., Vedral, V., Gu, M.: Converting coherence to quantum correlations. Phys. Rev. Lett. 116, 160407 (2016)

    Article  ADS  Google Scholar 

  8. Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)

    Article  ADS  Google Scholar 

  9. Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93, 012110 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  10. Chitambar, E., Streltsov, A., Rana, S., Bera, M.N., Adesso, G., Lewenstein, M.: Assisted distillation of quantum coherence. Phys. Rev. Lett. 116, 070402 (2016)

    Article  ADS  Google Scholar 

  11. Yu, C.S., Yang, S.R., Guo, B.Q.: Total quantum coherence and its applications. Quant. Inf. Process. 15, 3773 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  12. Chitambar, E., Gour, G.: Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett. 117, 030401 (2016)

    Article  ADS  Google Scholar 

  13. Napoli, C., Bromley, T.R., Cianciaruso, M., Piani, M., Johnston, N., Adesso, G.: Robustness of coherence: an operational and observable measure of quantum coherence. Phys. Rev. Lett. 116, 150502 (2016)

    Article  ADS  Google Scholar 

  14. Chitambar, E., Hsieh, M.H.: Relating the resource theories of entanglement and quantum coherence. Phys. Rev. Lett. 117, 020402 (2016)

    Article  ADS  Google Scholar 

  15. Marvian, I., Spekkens, R.W.: Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames. Phys. Rev. A 90, 062110 (2014)

    Article  ADS  Google Scholar 

  16. Marvian, I., Spekkens, R.W., Zanardi, P.: Quantum speed limits, coherence, and asymmetry. Phys. Rev. A 93, 052331 (2016)

    Article  ADS  Google Scholar 

  17. Singh, U., Zhang, L., Pati, A.K.: Average coherence and its typicality for random pure states. Phys. Rev. A 93, 032125 (2016)

    Article  ADS  Google Scholar 

  18. Piani, M., Cianciaruso, M., Bromley, T.R., Napoli, C., Johnston, N., Adesso, G.: Robustness of asymmetry and coherence of quantum states. Phys. Rev. A 93, 042107 (2016)

    Article  ADS  Google Scholar 

  19. Radhakrishnan, C., Parthasarathy, M., Jambulingam, S., Byrnes, T.: Distribution of quantum coherence in multipartite systems. Phys. Rev. Lett. 116, 150504 (2016)

    Article  ADS  Google Scholar 

  20. Bu, K.F., Singh, U., Fei, S.M., Pati, A.K., Wu, J.D.: Maximum relative entropy of coherence: an operational coherence measure. Phys. Rev. Lett. 119, 150405 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  21. Yuan, X., Zhou, H., Cao, Z., Ma, X.: Intrinsic randomness as a measure of quantum coherence. Phys. Rev. A 92, 022124 (2015)

    Article  ADS  Google Scholar 

  22. Qi, X., Gao, T., Yan, F.L.: Measuring coherence with entanglement concurrence. J. Phys. A Math. Theor. 50, 285301 (2017)

    Article  MathSciNet  Google Scholar 

  23. Du, S., Bai, S., Qi, X.: Coherence measures and optimal conversion for coherent states. Quantum Inf. Comput. 15, 1307 (2015)

    MathSciNet  Google Scholar 

  24. Bera, M.N., Qureshi, T., Siddiqui, M.A., Pati, A.K.: Duality of quantum coherence and path distinguishability. Phys. Rev. A 92, 012118 (2015)

    Article  ADS  Google Scholar 

  25. Bagan, E., Bergou, J.A., Cottrell, S.S., Hillery, M.: Relations between coherence and path information. Phys. Rev. Lett. 116, 160406 (2016)

    Article  ADS  Google Scholar 

  26. Bu, K.F., Li, L., Wu, J.D., Fei, S.M.: Duality relation between coherence and path information in the presence of quantum memory. J. Phys. A 51, 085304 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  27. Cheng, S., Hall, M.J.W.: Complementarity relations for quantum coherence. Phys. Rev. A 92, 042101 (2015)

    Article  ADS  Google Scholar 

  28. Singh, U., Bera, M.N., Dhar, H.S., Pati, A.K.: Maximally coherent mixed states: complementarity between maximal coherence and mixedness. Phys. Rev. A 91, 052115 (2015)

    Article  ADS  Google Scholar 

  29. Dai, Y., You, W., Dong, Y., Zhang, C.: Triangle inequalities in coherence measures and entanglement concurrence. Phys. Rev. A 96, 062308 (2017)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  32. Audenaert, K., Verstraete, F., De Moor, B.: Variational characterizations of separability and entanglement of formation. Phys. Rev. A 64, 052304 (2001)

    Article  ADS  Google Scholar 

  33. Rungta, P., Buzek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  34. Badziag, P., Deuar, P., Horodecki, M., Horodecki, P., Horodecki, R.: Concurrence in arbitrary dimensions. J. Mod. Opt. 49, 1289 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  35. Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  36. Albeverio, S., Fei, S.M.: A note on invariants and entanglements. J. Opt. B Quantum Semiclass Opt. 3, 223 (2001)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  38. Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: is there a bound entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  39. Horodeki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A. 232, 333 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  40. Dur, W., Cirac, J.I., Lewenstein, M., Bruß, D.: Distillability and partial transposition in bipartite systems. Phys. Rev. A. 61, 062313 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  41. Akhtarshenas, S.J.: Concurrence vectors in arbitrary multipartite quantum systems. J. Phys. A 38, 6777 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  42. Kim, J.S., Das, A., Sanders, B.S.: Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity. Phys. Rev. A. 79, 012329 (2009)

    Article  ADS  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments which helped to improve the results of the manuscript. This work is supported by the NSF of China under Grant No. 11675113 and NSF of Beijing under No. KZ201810028042.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Zhi-Xiang Jin & Shao-Ming Fei

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

    Xianqing Li-Jost & Shao-Ming Fei

  3. School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, China

    Xianqing Li-Jost

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  1. Zhi-Xiang Jin
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  2. Xianqing Li-Jost
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  3. Shao-Ming Fei
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Correspondence to Zhi-Xiang Jin.

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Jin, ZX., Li-Jost, X. & Fei, SM. Triangle-like inequalities related to coherence and entanglement negativity. Quantum Inf Process 18, 5 (2019). https://doi.org/10.1007/s11128-018-2121-5

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