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Quantum hypothesis testing and non-equilibrium statistical mechanics. (English) Zbl 1253.82052

The mathematical theory of non-equilibrium quantum statistical mechanics has developed rapidly in recent years. The research efforts are centered around the theory of entropic fluctuations and these developments are also concerned in this paper. As it is known there is a close interplay between information theory and statistical mechanics. One of the deepest links is provided by the theory of large deviations. In this paper, the authors attempt to interpret recent results in non-equilibrium statistical mechanics in terms of quantum information theory. They prove some results and elaborate the relations between quantum hypothesis testing and non-equilibrium statistical mechanics.
This paper consists of 8 sections. After an introduction in Section 2, they present a review with necessary proofs of results from large deviation theory. Then, Section 3 gives existing results in quantum hypothesis testing of finite quantum systems. After this, the description of non-equilibrium statistical mechanics of finite quantum systems with quantum hypothesis testing is discussed in Subsection 4.5. Section 5 presents results of modular theory. The next two sections are devoted to quantum hypothesis testing and non-equilibrium statistical mechanics of infinitely extended quantum systems described by \(W^*\)-algebras and \(W^*\)-dynamical systems. The last section describes several physical models for which the existence of the large deviation functionals is proven.

MSC:

82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C70 Transport processes in time-dependent statistical mechanics
46L53 Noncommutative probability and statistics
46N30 Applications of functional analysis in probability theory and statistics
62H15 Hypothesis testing in multivariate analysis
68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
81P50 Quantum state estimation, approximate cloning
81P68 Quantum computation
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References:

[1] DOI: 10.1007/s00023-006-0316-2 · Zbl 1115.82027
[2] DOI: 10.1007/s10955-006-9111-1 · Zbl 1197.82050
[3] DOI: 10.2977/prims/1195191148 · Zbl 0326.46031
[4] DOI: 10.2977/prims/1195190105 · Zbl 0374.46055
[5] DOI: 10.2977/prims/1195192744 · Zbl 0273.46054
[6] Araki H., Publ. Res. Inst. Math. Sci. Kyoto Univ. 18 pp 339–
[7] DOI: 10.1023/A:1024619726273 · Zbl 1032.82020
[8] DOI: 10.1007/3-540-33967-1_1 · Zbl 1126.82032
[9] DOI: 10.1063/1.2709849 · Zbl 1137.82331
[10] DOI: 10.1103/PhysRevLett.98.160501
[11] DOI: 10.1007/s00220-008-0417-5 · Zbl 1175.81036
[12] DOI: 10.1007/s00222-003-0318-3 · Zbl 1092.81043
[13] DOI: 10.1007/s00220-005-1426-2 · Zbl 1092.94013
[14] DOI: 10.1007/s00220-008-0440-6 · Zbl 1163.46041
[15] DOI: 10.1109/TIT.1974.1055254 · Zbl 0305.62017
[16] DOI: 10.1007/978-3-662-02520-8
[17] Bratteli O., Operator Algebras and Quantum Statistical Mechanics 2 (1996)
[18] DOI: 10.1016/0167-7152(93)90012-8 · Zbl 0797.60026
[19] DOI: 10.1214/aoms/1177729330 · Zbl 0048.11804
[20] Csiszár I., Studia Sci. Math. Hungar. 6 pp 181–
[21] DOI: 10.1007/BF01608389 · Zbl 0294.60080
[22] DOI: 10.1007/BF01011696
[23] DOI: 10.1007/978-1-4612-5320-4
[24] DOI: 10.1142/S0129055X03001679 · Zbl 1090.46049
[25] DOI: 10.1007/978-1-4613-8533-2
[26] DOI: 10.1142/9789812795106_0020
[27] DOI: 10.1007/s00023-003-0150-8 · Zbl 1106.82021
[28] DOI: 10.1088/0305-4470/35/50/307 · Zbl 1050.81007
[29] Hayashi M., Phys. Rev. A 76 pp 0623301–
[30] DOI: 10.1007/BF02100287 · Zbl 0756.46043
[31] DOI: 10.1063/1.2812417 · Zbl 1153.81375
[32] DOI: 10.1063/1.2872276 · Zbl 1153.81376
[33] Hiai F., J. Math. Phys. 49 pp 072104–
[34] DOI: 10.1214/aoms/1177700150 · Zbl 0135.19706
[35] Israel R., Convexity in the Theory of Lattice Gases (1979) · Zbl 0399.46055
[36] V. Jakšić, E. Kritchevski and C.A. Pillet, Large Coulomb Systems, Lecture Notes in Physics 695, eds. J. Dereziński and H. Siedentop (Springer, Berlin, 2006) pp. 141–211.
[37] DOI: 10.1007/s00220-006-0004-6 · Zbl 1104.82039
[38] DOI: 10.1007/s10955-006-9075-1 · Zbl 1101.82029
[39] DOI: 10.1007/s00220-006-0095-0 · Zbl 1147.82338
[40] DOI: 10.1007/s00023-007-0327-7 · Zbl 1375.82064
[41] Jakšić V., Comm. Math. Phys. 217 pp 285–
[42] DOI: 10.1023/A:1019818909696 · Zbl 1025.82011
[43] Jakšić V., Comm. Math. Phys. 226 pp 131–
[44] DOI: 10.1090/conm/447/08689
[45] DOI: 10.1088/0951-7715/24/3/003 · Zbl 1234.37012
[46] DOI: 10.1214/009053604000001219 · Zbl 1070.62117
[47] DOI: 10.1143/JPSJ.12.570
[48] DOI: 10.1007/978-3-642-58244-8 · Zbl 0996.60501
[49] Lebowitz J. L., Adv. Chem. Phys. 38 pp 109–
[50] Lehmann E. L., Testing Statistical Hypotheses (2005) · Zbl 1076.62018
[51] DOI: 10.1007/s10955-005-3015-3 · Zbl 1170.82307
[52] Levitov L. S., JETP Lett. 58 pp 230–
[53] DOI: 10.1063/1.1666274
[54] DOI: 10.1142/S0129055X03001850 · Zbl 1071.82535
[55] Merhav N., Foundations and Trends in Communications and Information Theory 6, in: Statistical Physics and Information Theory (2009) · Zbl 1211.94017
[56] DOI: 10.1007/s00023-007-0346-4 · Zbl 1193.82023
[57] DOI: 10.1016/j.jfa.2006.10.017 · Zbl 1122.81043
[58] DOI: 10.1063/1.3085759 · Zbl 1187.82012
[59] DOI: 10.1063/1.2953473 · Zbl 1152.81567
[60] DOI: 10.1142/9789812563071_0011
[61] DOI: 10.1007/s10955-004-3452-4 · Zbl 1113.82008
[62] DOI: 10.1063/1.3451110 · Zbl 1311.81069
[63] DOI: 10.1214/08-AOS593 · Zbl 1162.62100
[64] DOI: 10.1007/s00220-003-1011-5 · Zbl 1092.82026
[65] DOI: 10.1007/s00220-010-0986-y · Zbl 1193.82007
[66] DOI: 10.1007/s11005-011-0504-y · Zbl 1269.46038
[67] DOI: 10.1109/TIT.2004.828155 · Zbl 1303.81070
[68] Ogawa T., IEEE Trans. Inform. Theory 46 pp 2428–
[69] Ohya M., Quantum Entropy and Its Use (2004)
[70] Pillet C.-A., Markov Process. Related Fields 7 pp 145–
[71] Rockafellar R. T., Convex Analysis (1972) · Zbl 0224.49003
[72] DOI: 10.1142/S0129055X09003694 · Zbl 1173.82338
[73] DOI: 10.1007/s002200100534 · Zbl 1051.82003
[74] DOI: 10.1142/S0129055X02001296 · Zbl 1027.82027
[75] DOI: 10.1515/9781400863433
[76] Strǎtilǎ Ş., Modular Theory in Operator Algebras (1981)
[77] DOI: 10.1007/978-1-4612-6188-9
[78] DOI: 10.1142/9789812774835_0014
[79] DOI: 10.1142/9789812704412_0006
[80] T. Matsui and S. Tasaki, Stochastic Analysis: Classical and Quantum, ed. T. Hida (World Scientific, Singapore, 2005) pp. 211–227.
[81] DOI: 10.1007/BF01609834 · Zbl 0358.46026
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