Solutions for the MaxEnt problem with symmetry constraints
Tóm tắt
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.
Tài liệu tham khảo
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957)
Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171 (1957)
Katz, A.: Principles of Statistical Mechanics—The Information Theory Approach. Freeman, San Francisco (1967)
Pressé, S., Ghosh, K., Lee, J., Dill, K.: Principles of maximum entropy and maximum caliber in statistical physics. Rev. Mod. Phys. 85, 1115 (2013)
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)
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)
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)
Diambra, L., Plastino, A.: Maximum entropy, pseudoinverse techniques, and time series predictions with layered networks. Phys. Rev. E 52, 4557 (1995)
Rebollo-Neira, L., Plastino, A.: Nonextensive maximum-entropy-based formalism for data subset selection. Phys. Rev. E 65, 011113 (2001)
Diambra, L., Plastino, A.: Maximum-entropy principle and neural networks that learn to construct approximate wave functions. Phys. Rev. E 53, 1021 (1996)
Goswami, G., Prasad, J.: Maximum entropy deconvolution of primordial power spectrum. Phys. Rev. D 88, 023522 (2013)
Cofré, R., Cessac, B.: Exact computation of the maximum-entropy potential of spiking neural-network models. Phys. Rev. E 89, 052117 (2014)
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)
Canosa, N., Plastino, A., Rossignoli, R.: Ground-state wave functions and maximum entropy. Phys. Rev. A 40, 519 (1989)
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)
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)
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)
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)
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)
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)
Buzĕk, V., Drobný, G.: Quantum tomography via the MaxEnt principle. J. Mod. Opt. 47(14/15), 2823–2839 (2000)
Ziman, M.: Incomplete quantum process tomography and principle of maximal entropy. Phys. Rev. A 78, 032118 (2008)
Holik, F., Plastino, A.: Quantal effects and MaxEnt. J. Math. Phys. 53, 073301 (2012)
Holik, F., Massri, C., Plastino, A.: Geometric probability theory and Jaynes’s methodology. Int. J. Geom. Methods Mod. Phys. 13, 1650025 (2016)
Paris, M., Rehacek, J.: Quantum State Estimation. Springer, New York (2010)
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)
Holik, F., Bosyk, G.M., Bellomo, G.: Quantum information as a non-Kolmogorovian generalization of Shannon’s theory. Entropy 17, 7349–7373 (2015)
Holik, F., Plastino, A., Sáenz, M.: Natural information measures for contextual probabilistic models. Quantum Inf. Comput. 16, 87–104 (2016)
Rényi, A.: On measures of entropy and information. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547–561 (1961)
Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52(1–2), 479–487 (1988)
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)
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)
Rastegin, A.E.: Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis’ entropies. Quantum Inf. Process. 12(9), 2947–2963 (2013)
Zhang, J., Zhang, Y., Yu, C.: Rényi entropy uncertainty relation for successive projective measurements. Quantum Inf. Process. 14(6), 2239–2253 (2015)
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)
Kurzyk, D., Pawela, Ł., Puchała, Z.: Unconditional security of a \(K\)-state quantum key distribution protocol. Quantum Inf. Process. 17, 228 (2018)
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)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)