Hilbert–Schmidt quantum coherence in multi-qudit systems

  • Jonas Maziero


Using Bloch’s parametrization for qudits (d-level quantum systems), we write the Hilbert–Schmidt distance (HSD) between two generic n-qudit states as an Euclidean distance between two vectors of observables mean values in \(\mathbb {R}^{\Pi _{s=1}^{n}d_{s}^{2}-1}\), where \(d_{s}\) is the dimension for qudit s. Then, applying the generalized Gell–Mann’s matrices to generate \(\mathrm{SU}(d_{s})\), we use that result to obtain the Hilbert–Schmidt quantum coherence (HSC) of n-qudit systems. As examples, we consider in detail one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied, and the contrasting behaviors of HSC, \(l_{1}\)-norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.


Quantum coherence Hilbert–Schmidt distance Qudit states Gell–Mann matrices 



This work was supported by the Brazilian funding agencies: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), processes 441875/2014-9 and 303496/2014-2, Coordenação de Desenvolvimento de Pessoal de Nível Superior (CAPES), process 6531/2014-08, and Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 2008/57856-6.


  1. 1.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 3. Addison-Wesley, Boston (1965)zbMATHGoogle Scholar
  2. 2.
    Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (2008)Google Scholar
  3. 3.
    Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ma, J., Yadin, B., Girolami, D., Vedral, V., Gu, M.: Converting coherence to quantum correlations. Phys. Rev. Lett. 116, 160407 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Giorda, P., Allegra, M.: Coherence in quantum estimation. arXiv:1611.02519
  6. 6.
    Chenu, A., Scholes, G.D.: Coherence in energy transfer and photosynthesis. Annu. Rev. Phys. Chem. 66, 69 (2015)ADSCrossRefGoogle Scholar
  7. 7.
    Uzdin, R.: Coherence-induced reversibility and collective operation of quantum heat machines via coherence recycling. Phys. Rev. Appl. 6, 024004 (2016)ADSCrossRefGoogle Scholar
  8. 8.
    Hillery, M.: Coherence as a resource in decision problems: The Deutsch–Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Yuan, X., Liu, K., Xu, Y., Wang, W., Ma, Y., Zhang, F., Yan, Z., Vijay, R., Sun, L., Ma, X.: Experimental quantum randomness processing using superconducting qubits. Phys. Rev. Lett. 117, 010502 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    Streltsov, A., Chitambar, E., Rana, S., Bera, M.N., Winter, A., Lewenstein, M.: Entanglement and coherence in quantum state merging. Phys. Rev. Lett. 116, 240405 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Misra, A., Singh, U., Bhattacharya, S., Pati, A.K.: Energy cost of creating quantum coherence. Phys. Rev. A 93, 052335 (2016)ADSCrossRefGoogle Scholar
  12. 12.
    Shi, H.-L., Liu, S.-Y., Wang, X.-H., Yang, W.-L., Yang, Z.-Y., Fan, H.: Coherence depletion in the Grover quantum search algorithm. Phys. Rev. A 95, 032307 (2017)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Streltsov, A., Adesso, G., Plenio, M.B.: Quantum coherence as a resource. arXiv:1609.02439
  14. 14.
    Chitambar, E., Gour, G.: Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett. 117, 030401 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Marvian, I., Spekkens, R.W.: How to quantify coherence: distinguishing speakable and unspeakable notions. Phys. Rev. A 94, 052324 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 170401 (2014)CrossRefGoogle Scholar
  17. 17.
    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)ADSCrossRefGoogle Scholar
  18. 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)ADSCrossRefGoogle Scholar
  19. 19.
    Levi, F., Mintert, F.: A quantitative theory of coherent delocalization. New J. Phys. 16, 033007 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Pérez-García, D., Wolf, M.M., Petz, D., Ruskai, M.B.: Contractivity of positive and trace-preserving maps under Lp norms. J. Math. Phys. 47, 083506 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Wang, X., Schirmer, S.G.: Contractivity of the Hilbert–Schmidt distance under open-system dynamics. Phys. Rev. A 79, 052326 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Dajka, J., Łuczka, J., Hänggi, P.: Distance between quantum states in the presence of initial qubit-environment correlations: a comparative study. Phys. Rev. A 84, 032120 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    Ozawa, M.: Entanglement measures and the Hilbert–Schmidt distance. Phys. Lett. A 268, 158 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Piani, M.: The problem with the geometric discord. Phys. Rev. A 86, 034101 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    Dakic, B., Lipp, Y.O., Ma, X., Ringbauer, M., Kropatschek, S., Barz, S., Paterek, T., Vedral, V., Zeilinger, A., Brukner, C., Walther, P.: Quantum discord as resource for remote state preparation. Nat. Phys. 8, 666 (2012)CrossRefGoogle Scholar
  26. 26.
    Wu, X., Zhou, T.: Geometric discord: a resource for increments of quantum key generation through twirling. Sci. Rep. 5, 13365 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Bertlmann, R.A., Narnhofer, H., Thirring, W.: A geometric picture of entanglement and Bell inequalities. Phys. Rev. A 66, 032319 (2002)ADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    Bertlmann, R.A., Durstberger, K., Hiesmayr, B.C., Krammer, P.: Optimal entanglement witnesses for qubits and qutrits. Phys. Rev. A 72, 052331 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Lee, J., Kim, M.S., Brukner, C.: Operationally invariant measure of the distance between quantum states by complementary measurements. Phys. Rev. Lett. 91, 087902 (2003)ADSCrossRefGoogle Scholar
  30. 30.
    Tamir, B., Cohen, E.: A Holevo-type bound for a Hilbert Schmidt distance measure. J. Quantum Inf. Sci. 05, 127 (2015)CrossRefGoogle Scholar
  31. 31.
    Dodonov, V.V., Man’ko, O.V., Man’ ko, V.I., Wünsche, A.: Hilbert–Schmidt distance and non-classicality of states in quantum optics. J. Mod. Opt. 47, 633 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Zyczkowski, K., Sommers, H.-J.: Hilbert–Schmidt volume of the set of mixed quantum states. J. Phys. A Math. Gen. 36, 10115 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754 (2006)CrossRefGoogle Scholar
  34. 34.
    Björk, G., de Guise, H., Klimov, A.B., de la Hoz, P., Sánchez-Soto, L.L.: Classical distinguishability as an operational measure of polarization. Phys. Rev. A 90, 013830 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    Witte, C., Trucks, M.: A new entanglement measure induced by the Hilbert–Schmidt norm. Phys. Lett. A 257, 14 (1999)ADSCrossRefGoogle Scholar
  36. 36.
    Verstraete, F., Dehaene, J., De Moor, B.: On the geometry of entangled states. J. Mod. Opt. 49, 1277 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Maziero, J.: Computing partial transposes and related entanglement functions. Braz. J. Phys. 46, 605 (2016)ADSGoogle Scholar
  38. 38.
    Dakic, B., Vedral, V., Brukner, C.: Necessary and sufficient condition for non-zero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Li, L., Wang, Q.-W., Shen, S.-Q., Li, M.: Geometric measure of quantum discord with weak measurements. Quantum Inf. Process. 15, 291 (2016)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Luo, S., Fu, S.: Geometric measure of quantum discord. Phys. Rev. A 82, 034302 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Akhtarshenas, S.J., Mohammadi, H., Karimi, S., Azmi, Z.: Computable measure of quantum correlation. Quantum Inf. Process. 14, 247 (2015)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Luo, S., Fu, S.: Measurement-induced nonlocality. Phys. Rev. Lett. 106, 120401 (2011)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Girolami, D., Adesso, G.: Interplay between computable measures of entanglement and other quantum correlations. Phys. Rev. A 84, 052110 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    Yuan, Y.-L., Hou, X.-W.: Thermal geometric discords in a two-qutrit system. Int. J. Quantum Inf. 14, 1650016 (2016)CrossRefzbMATHGoogle Scholar
  45. 45.
    Pozzobom, M.B., Maziero, J.: Environment-induced quantum coherence spreading of a qubit. Ann. Phys. 377, 243 (2017)ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Chandrashekar, R., Manikandan, P., Segar, J., Byrnes, T.: Distribution of quantum coherence in multipartite systems. Phys. Rev. Lett. 116, 150504 (2016)CrossRefGoogle Scholar
  47. 47.
    Tan, K.C., Kwon, H., Park, C.-Y., Jeong, H.: Unified view of quantum correlations and quantum coherence. Phys. Rev. A 94, 022329 (2016)ADSCrossRefGoogle Scholar
  48. 48.
    Audenaert, K.M.R., Calsamiglia, J., Munõz-Tapia, R., Bagan, E., Masanes, L., Acin, A., Verstraete, F.: Discriminating states: the quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007)ADSCrossRefGoogle Scholar
  49. 49.
    Maziero, J.: Non-monotonicity of trace distance under tensor products. Braz. J. Phys. 45, 560 (2015)ADSCrossRefGoogle Scholar
  50. 50.
    Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93, 012110 (2016)ADSCrossRefMathSciNetGoogle Scholar
  51. 51.
    Shao, L.-H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2015)ADSCrossRefGoogle Scholar
  52. 52.
    Wang, Z., Wang, Y.-L., Wang, Z.-X.: Trace distance measure of coherence for a class of qudit states. Quantum Inf. Process. 15, 4641 (2016)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Kimura, G.: The Bloch vector for N-level systems. Phys. Lett. A 314, 339 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Bertlmann, R.A., Krammer, P.: Bloch vectors for qudits. J. Phys. A Math. Theor. 41, 235303 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Maziero, J.: Computing coherence vectors and correlation matrices with application to quantum discord quantification. Adv. Math. Phys. 2016, e6892178 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
  57. 57.
    Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)ADSCrossRefGoogle Scholar
  58. 58.
    Bromley, T.R., Cianciaruso, M., Adesso, G.: Frozen quantum coherence. Phys. Rev. Lett. 114, 210401 (2015)ADSCrossRefGoogle Scholar
  59. 59.
    Guo, J.-L., Li, H., Long, G.-L.: Decoherent dynamics of quantum correlations in qubit-qutrit systems. Quantum Inf. Process. 12, 3421 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Khan, S., Ahmad, I.: Environment generated quantum correlations in bipartite qubit–qutrit systems. Optik 127, 2448 (2016)ADSCrossRefGoogle Scholar
  61. 61.
    Soares-Pinto, D.O., Moussa, M.H.Y., Maziero, J., de Azevedo, E.R., Bonagamba, T.J., Serra, R.M., Céleri, L.C.: Equivalence between Redfield- and master-equation approaches for a time-dependent quantum system and coherence control. Phys. Rev. A 83, 062336 (2011)ADSCrossRefGoogle Scholar
  62. 62.
    Maziero, J.: Fortran code for generating random probability vectors, unitaries, and quantum states. Front. ICT 3, 4 (2016)CrossRefGoogle Scholar
  63. 63.

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Departamento de Física, Centro de Ciências Naturais e ExatasUniversidade Federal de Santa MariaSanta MariaBrazil

Personalised recommendations