Coherence measures based on coherence eigenvalue and their applications

  • Lei LiEmail author
  • Qing-Wen Wang
  • Shu-Qian Shen
  • Ming Li


The quantification of coherence is central in quantum information theory and far from being fully understood, particularly for multipartite quantum states. In this work, we first propose the coherence eigenvalue for multipartite pure quantum states and then define the coherence measure for a given multipartite pure or mixed state based on it. It is found that the convex roof coherence measure is equal to the geometric measure based on fidelity. This finding allows us to give an alternative way to prove that the geometric measure of coherence is monotone under selective measurements on average. The other two new coherence measures based on coherence eigenvalue are presented, and their properties are also investigated. As an application, the bounds of several well-known coherence measures are derived by using of coherence eigenvalue. In particular, by proposing a new coherence witness which is related to coherence eigenvalue, we obtain a bound to robustness of coherence and give it a corresponding geometric interpretation.



The authors would like to thank the editor and anonymous referees for their valuable suggestions. This research was supported by the Fundamental Research Funds for the Central Universities No. 18CX02023A, 18CX02035A, the Grants from the NSFC (11775306, 11571220), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission(13ZZ080).


  1. 1.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)CrossRefADSzbMATHGoogle Scholar
  2. 2.
    Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nat. Photon. 5, 222 (2011)CrossRefADSGoogle Scholar
  3. 3.
    Tóth, G., Apellaniz, I.: Quantum metrology from a quantum information science perspective. J. Phys. A Math. Theor. 47, 424006 (2014)CrossRefADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Girolami, D., Souza, A.M., Giovannetti, V., Tufarelli, T., Filgueiras, J.G., Sarthour, R.S., Soares-Pinto, D.O., Oliveira, I.S., Adesso, G.: Quantum discord determines the interferometric power of quantum states. Phys. Rev. Lett. 112, 210401 (2014)CrossRefADSGoogle Scholar
  5. 5.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  6. 6.
    Horodecki, M., Oppenheim, J.: Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013)CrossRefADSGoogle Scholar
  7. 7.
    Narasimhachar, V., Gour, G.: Low-temperature thermodynamics with quantum coherence. Nat. Commun. 6, 7689 (2015)CrossRefADSGoogle Scholar
  8. 8.
    Gour, G., Spekkens, R.W.: The resource theory of quantum reference frames: manipulations and monotones. New J. Phys. 10, 033023 (2008)CrossRefADSGoogle Scholar
  9. 9.
    Marvian, I., Spekkens, R.W.: Asymmetry properties of pure quantum states. Phys. Rev. A 90, 062110 (2014)CrossRefADSGoogle Scholar
  10. 10.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)CrossRefADSGoogle Scholar
  11. 11.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)CrossRefADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)CrossRefADSGoogle Scholar
  14. 14.
    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)CrossRefADSGoogle Scholar
  15. 15.
    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)CrossRefADSGoogle Scholar
  16. 16.
    Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93, 012110 (2016)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Shao, L., Xi, Z., Fan, H., Li, Y.: The fidelity and trace norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2015)CrossRefADSGoogle Scholar
  18. 18.
    Bu, K., Singh, U., Fei, S., Pati, A.K., Wu, J.: Maximum relative entropy of coherence: an operational coherence measure. Phys. Rev. Lett. 119, 150405 (2017)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Chitambar, E., Streltsov, A., Rana, S., Bera, M.N., Adesso, G., Lewenstein, M.: Assisted distillation of quantum coherence. Phys. Rev. Lett. 116, 070402 (2016)CrossRefADSGoogle Scholar
  20. 20.
    Yu, C.: Quantum coherence via skew information and its polygamy. Phys. Rev. A 95, 042337 (2017)CrossRefADSGoogle Scholar
  21. 21.
    Luo, S., Sun, Y.: Partial coherence with application to the monotonicity problem of coherence involving skew information. Phys. Rev. A 96, 022136 (2017)CrossRefADSGoogle Scholar
  22. 22.
    Girolami, D.: Observable measure of quantum coherence in finite dimensional systems. Phys. Rev. Lett. 113, 170401 (2014)CrossRefADSGoogle Scholar
  23. 23.
    Yuan, X., Zhou, H., Cao, Z., Ma, X.: Intrinsic randomness as a measure of quantum coherence. Phys. Rev. A 92, 022124 (2015)CrossRefADSGoogle Scholar
  24. 24.
    Rastegin, A.E.: Quantum-coherence quantifiers based on the Tsallis relative \(\alpha \)-entropies. Phys. Rev. A 93, 032136 (2016)CrossRefADSGoogle Scholar
  25. 25.
    Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Guo, Y., Goswami, S.: Discordlike correlation of bipartite coherence. Phys. Rev. A 95, 062340 (2017)CrossRefADSGoogle Scholar
  27. 27.
    Yao, Y., Xiao, X., Ge, L., Sun, C.P.: Quantum coherence in multipartite systems. Phys. Rev. A 92, 022112 (2015)CrossRefADSGoogle Scholar
  28. 28.
    Xi, Z., Li, Y., Fan, H.: Quantum coherence and correlations in quantum system. Sci. Rep. 5, 10922 (2015)CrossRefADSGoogle Scholar
  29. 29.
    Ma, J., Yadin, B., Girolami, D., Vedral, V., Gu, M.: Converting coherence to quantum correlations. Phys. Rev. Lett. 116, 160407 (2016)CrossRefADSGoogle Scholar
  30. 30.
    Zhu, H., Ma, Z., Cao, Z., Fei, S., Vedral, V.: Operational one-to-one mapping between coherence and entanglement measures. Phys. Rev. A 96, 032316 (2017)CrossRefADSGoogle Scholar
  31. 31.
    Tan, K.C., Kwon, H., Park, C.Y., Jeong, H.: A unified view of quantum correlations and quantum coherence. Phys. Rev. A 94, 022329 (2016)CrossRefADSGoogle Scholar
  32. 32.
    Wei, T.C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)CrossRefADSGoogle Scholar
  33. 33.
    Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009) CrossRefADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Waldraff, F.B., Braun, D., Giraud, O.: Partial transpose criteria for symmetric states. Phys. Rev. A 94, 042324 (2016)CrossRefADSGoogle Scholar
  35. 35.
    Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331, 593–622 (2014)CrossRefADSzbMATHGoogle Scholar
  36. 36.
    Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013)CrossRefADSMathSciNetzbMATHGoogle Scholar
  37. 37.
    Gühne, O., Reimpell, M., Werner, R.F.: Estimating entanglement measures in experiments. Phys. Rev. Lett. 98, 110502 (2007)CrossRefADSGoogle Scholar
  38. 38.
    Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)CrossRefADSGoogle Scholar
  39. 39.
    Tóth, G., Gühne, O.: Detecting genuine multipartite entanglement with two local measurements. Phys. Rev. Lett. 94, 060501 (2005)CrossRefADSGoogle Scholar
  40. 40.
    Tóth, G., Knapp, C., Gühne, O., Briegel, H.J.: Optimal spin squeezing inequalities detect bound entanglement in spin models. Phys. Rev. Lett. 99, 250405 (2007)CrossRefADSGoogle Scholar
  41. 41.
    Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.: Quantifying entanglement. Phys. Rev. Lett. 78, 2275 (1997) CrossRefADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Kiesel, N., Schmid, C., Weber, U., Töth, G., Gühne, O., Ursin, R., Weinfurter, H.: Experimental analysis of a four-qubit photon cluster state. Phys. Rev. Lett. 95, 210502 (2005)CrossRefADSGoogle Scholar
  43. 43.
    Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states. Phys. Rev. A 77, 012104 (2008)CrossRefADSGoogle Scholar
  44. 44.
    Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication. Phys. Rev. Lett. 96, 040501 (2006)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Tamaryan, L., Park, D.K., Tamaryan, S.: Analytic expressions for geometric measure of three qubit states. Phys. Rev. A 77, 022325 (2008)CrossRefADSGoogle Scholar
  46. 46.
    Hilling, J.J., Sudbery, A.: The geometric measure of multipartite entanglement and the singular values of a hypermatrix. J. Math. Phys. 91, 072102 (2010)CrossRefADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Chen, X.Y.: Entanglement of graph states up to eight qubits. J. Phys. B At. Mol. Opt. Phys. 43, 085507 (2010)CrossRefADSGoogle Scholar
  48. 48.
    Hubener, R., Kleinmann, M., Wei, T.C., Guillen, C.G., Gühne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009)CrossRefADSMathSciNetzbMATHGoogle Scholar
  50. 50.
    Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93, 012304 (2016)CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Hayashi, M.: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin (2017)CrossRefGoogle Scholar
  52. 52.
    Mosonyi, M., Ogawa, T.: Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Commun. Math. Phys. 334, 1617–1648 (2015)CrossRefADSzbMATHGoogle Scholar
  53. 53.
    Konig, R., Renner, R., Schaffner, C.: The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55, 4337–4347 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Tomamichel, M., Berta, M., Hayashi, M.: Relating different quantum generalizations of the conditional Rényi entropy. J. Math. Phys. 55, 082206 (2014)CrossRefADSMathSciNetzbMATHGoogle Scholar
  55. 55.
    Zhu, H., Hayashi, M., Chen, L.: Coherence and entanglement measures based on Rényi relative entropies. J. Phys. A Math. Theor. 50, 475303 (2017)CrossRefADSGoogle Scholar
  56. 56.
    Brandão, F.G.S.L.: Quantifying entanglement with witness operators. Phys. Rev. A 72, 022310 (2005)CrossRefADSMathSciNetGoogle Scholar
  57. 57.
    Streltsov, A., Kampermann, H., Bru\(\beta \), D.: Linking a distance measure of entanglment to its convex roof. New J. Phys. 12, 123004 (2010)Google Scholar
  58. 58.
    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)CrossRefADSGoogle Scholar
  59. 59.
    Orús, R.: Universal geometric entanglement close to quantum phase transitions. Phys. Rev. Lett. 100, 130502 (2008)CrossRefADSGoogle Scholar
  60. 60.
    Orús, R., Dusuel, S., Vidal, J.: Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model. Phys. Rev. Lett. 101, 025701 (2008)CrossRefADSGoogle Scholar
  61. 61.
    Orús, R.: Geometric entanglement in a one-dimensional valence bond solid state. Phys. Rev. A 78, 062332 (2008)CrossRefADSGoogle Scholar

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Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoPeople’s Republic of China
  2. 2.College of ScienceShanghai UniversityShanghaiPeople’s Republic of China

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