Quantum Information Processing

, 18:293 | Cite as

Solutions for the MaxEnt problem with symmetry constraints

  • Marcelo Losada
  • Federico HolikEmail author
  • Cesar Massri
  • Angelo Plastino


In this paper, we deal with the situation in which the unknown state of a quantum system has to be estimated under the assumption that it is prepared obeying a known set of symmetries. We present a system of equations and an explicit solution for the problem of determining the MaxEnt state satisfying these constraints. Our approach can be applied to very general situations, including symmetries of the source represented by Lie and finite groups.


Maximum entropy principle Symmetries in quantum mechanics Quantum state estimation 



This research was funded by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).


  1. 1.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171 (1957)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Katz, A.: Principles of Statistical Mechanics—The Information Theory Approach. Freeman, San Francisco (1967)Google Scholar
  4. 4.
    Pressé, S., Ghosh, K., Lee, J., Dill, K.: Principles of maximum entropy and maximum caliber in statistical physics. Rev. Mod. Phys. 85, 1115 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    Cavagna, A., Giardina, I., Ginelli, F., Mora, T., Piovani, D., Tavarone, R., Walczak, A.M.: Dynamical maximum entropy approach to flocking. Phys. Rev. E 89, 042707 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    Beretta, G.P.: Steepest entropy ascent model for far-nonequilibrium thermodynamics: unified implementation of the maximum entropy production principle. Phys. Rev. E 90, 042113 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Sinatra, R., Gómez-Gardeñes, J., Lambiotte, R., Nicosia, V., Latora, V.: Maximal-entropy random walks in complex networks with limited information. Phys. Rev. E 83, 030103 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    Diambra, L., Plastino, A.: Maximum entropy, pseudoinverse techniques, and time series predictions with layered networks. Phys. Rev. E 52, 4557 (1995)ADSCrossRefGoogle Scholar
  9. 9.
    Rebollo-Neira, L., Plastino, A.: Nonextensive maximum-entropy-based formalism for data subset selection. Phys. Rev. E 65, 011113 (2001)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Diambra, L., Plastino, A.: Maximum-entropy principle and neural networks that learn to construct approximate wave functions. Phys. Rev. E 53, 1021 (1996)ADSCrossRefGoogle Scholar
  11. 11.
    Goswami, G., Prasad, J.: Maximum entropy deconvolution of primordial power spectrum. Phys. Rev. D 88, 023522 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Cofré, R., Cessac, B.: Exact computation of the maximum-entropy potential of spiking neural-network models. Phys. Rev. E 89, 052117 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Lanatà, N., Strand, H.U.R., Yao, Y., Kotliar, G.: Principle of maximum entanglement entropy and local physics of strongly correlated materials. Phys. Rev. Lett. 113, 036402 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Canosa, N., Plastino, A., Rossignoli, R.: Ground-state wave functions and maximum entropy. Phys. Rev. A 40, 519 (1989)ADSCrossRefGoogle Scholar
  15. 15.
    Arrachea, L., Canosa, N., Plastino, A., Portesi, M., Rossignoli, R.: Maximum-entropy approach to critical phenomena in ground states of finite systems. Phys. Rev. A 45, 7104 (1992)ADSCrossRefGoogle Scholar
  16. 16.
    Tkačik, G., Marre, O., Mora, T., Amodei, D., Berry II, M.J., Bialek, W.: The simplest maximum entropy model for collective behavior in a neural network. J. Stat. Mech. Theory Exp. 2013, P03011 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gonçalves, D., Lavor, C., Gomes-Ruggiero, M., Cesário, A., Vianna, R., Maciel, T.: Quantum state tomography with incomplete data: maximum entropy and variational quantum tomography. Phys. Rev. A 87, 052140 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Gzyl, H., ter Horst, E., Molina, G.: Application of the method of maximum entropy in the mean to classification problems. Phys. A Stat. Mech. Appl. 437, 101–108 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Alves, A., Dias, A.G., da Silva, R.: Maximum entropy principle and the Higgs boson mass. Phys. A Stat. Mech. Appl. 420, 1–7 (2015)CrossRefGoogle Scholar
  20. 20.
    Schönfeldt, J.-H., Jimenez, N., Plastino, A.R., Plastino, A.L., Casas, M.: Maximum entropy principle and classical evolution equations with source terms. Phys. A Stat. Mech. Appl. 374, 573–584 (2007)CrossRefGoogle Scholar
  21. 21.
    Buzĕk, V., Drobný, G.: Quantum tomography via the MaxEnt principle. J. Mod. Opt. 47(14/15), 2823–2839 (2000)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ziman, M.: Incomplete quantum process tomography and principle of maximal entropy. Phys. Rev. A 78, 032118 (2008)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Holik, F., Plastino, A.: Quantal effects and MaxEnt. J. Math. Phys. 53, 073301 (2012)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Holik, F., Massri, C., Plastino, A.: Geometric probability theory and Jaynes’s methodology. Int. J. Geom. Methods Mod. Phys. 13, 1650025 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Paris, M., Rehacek, J.: Quantum State Estimation. Springer, New York (2010)zbMATHGoogle Scholar
  26. 26.
    Stefano, Q.P., Rebón, L., Ledesma, S., Iemmi, C.: Determination of any pure spatial qudits from a minimum number of measurements by phase-stepping interferometry. Phys. Rev. A 96, 062328 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    Holik, F., Bosyk, G.M., Bellomo, G.: Quantum information as a non-Kolmogorovian generalization of Shannon’s theory. Entropy 17, 7349–7373 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    Holik, F., Plastino, A., Sáenz, M.: Natural information measures for contextual probabilistic models. Quantum Inf. Comput. 16, 87–104 (2016)MathSciNetGoogle Scholar
  29. 29.
    Rényi, A.: On measures of entropy and information. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547–561 (1961)Google Scholar
  30. 30.
    Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52(1–2), 479–487 (1988)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Bosyk, G.M., Zozor, S., Holik, F., Portesi, M., Lamberti, P.W.: A family of generalized quantum entropies: definition and properties. Quantum Inf. Process. 15, 3393–3420 (2016)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Portesi, M., Holik, F., Lamberti, P.W., Bosyk, G.M., Bellomo, G., Zozor, S.: Generalized entropies in quantum and classical statistical theories. Eur. Phys. J. Spec. Top. 227(3–4), 335–344 (2018)CrossRefGoogle Scholar
  33. 33.
    Rastegin, A.E.: Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis’ entropies. Quantum Inf. Process. 12(9), 2947–2963 (2013)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, J., Zhang, Y., Yu, C.: Rényi entropy uncertainty relation for successive projective measurements. Quantum Inf. Process. 14(6), 2239–2253 (2015)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Khosravi Tanak, A., Mohtashami Borzadaran, G.R., Ahmadi, J.: Maximum Tsallis entropy with generalized Gini and Gini mean difference indices constraints. Phys. A Stat. Mech. Appl. 471, 554–560 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kurzyk, D., Pawela, Ł., Puchała, Z.: Unconditional security of a \(K\)-state quantum key distribution protocol. Quantum Inf. Process. 17, 228 (2018)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Liang, Y., Feng, X., Chen, W.: Monogamy and polygamy for multi-qubit W-class states using convex-roof extended negativity of assistance and Rényi-\(\alpha \) entropy. Quantum Inf. Process. 16, 300 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)zbMATHGoogle Scholar
  39. 39.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Marcelo Losada
    • 1
  • Federico Holik
    • 2
    • 3
    • 4
    Email author
  • Cesar Massri
    • 4
  • Angelo Plastino
    • 3
  1. 1.Universidad de Buenos Aires - CONICETCiudad de Buenos AiresArgentina
  2. 2.Center Leo Apostel for Interdisciplinary Studies and, Department of MathematicsBrussels Free UniversityBrusselsBelgium
  3. 3.National University La Plata - CONICET IFLP-CCTLa PlataArgentina
  4. 4.Department of MathematicsUniversity CAECE - CONICET IMASBuenos AiresArgentina

Personalised recommendations