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Asymptotic and non-asymptotic analysis for a hidden Markovian process with a quantum hidden system. (English) Zbl 1397.81026

Summary: We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data. Using these bounds, we derive the central limit theorem and large and moderate deviations for the tail probability. Then, we give the asymptotic variance by using the second derivative of the cumulant generating function. We also derive another expression for the asymptotic variance by considering the quantum version of the fundamental matrix. Further, we explain how to extend our results to a general probabilistic system.

MSC:

81P15 Quantum measurement theory, state operations, state preparations
62J10 Analysis of variance and covariance (ANOVA)
62G32 Statistics of extreme values; tail inference
60F10 Large deviations
60F05 Central limit and other weak theorems
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[1] Herrero-Collantes, M.; Garcia-Escartin, J. C., Quantum random number generators, Rev. Mod. Phys., 89, (2017) · doi:10.1103/RevModPhys.89.015004
[2] Caruso, F.; Giovannetti, V.; Lupo, C.; Mancini, S., Quantum channels and memory effects, Rev. Mod. Phys., 86, 1203, (2014) · doi:10.1103/RevModPhys.86.1203
[3] Cao, M. X.; Vontobel, P. O., Estimating the information rate of a channel with classical input and output and a quantum state, (2017)
[4] Kretchmann, D.; Werner, R. F., Quantum channels with memory, Phys. Rev. A, 72, (2005) · doi:10.1103/PhysRevA.72.062323
[5] Ozawa, M., Quantum measuring processes of continuous observables, J. Math. Phys., 25, 79, (1984) · doi:10.1063/1.526000
[6] Kontoyiannis, I.; Meyn, S. P., Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann. Appl. Probab., 13, 304-362, (2003) · Zbl 1016.60066 · doi:10.1214/aoap/1042765670
[7] Meyn, S. P.; Tweedie, R. L., Markov Chains and Stochastic Stability, (1993), London: Springer, London · Zbl 0925.60001
[8] Jones, G. L., On the Markov chain central limit theorem, Probab. Surv., 1, 299-320, (2004) · Zbl 1189.60129 · doi:10.1214/154957804100000051
[9] Ben-Ari, I.; Neumann, M., Probabilistic approach to perron root, the group inverse, and applications, Linear Multilinear Algebr., 60, 39-63, (2012) · Zbl 1239.15008 · doi:10.1080/03081087.2011.559167
[10] Donsker, M. D.; Varadhan, S. R S.; Donsker, M. D.; Varadhan, S. R S., Asymptotic evaluation of certain Markov process expectations for large time, I, II. Asymptotic evaluation of certain Markov process expectations for large time, I, II, Commun. Pure Appl. Math.. Commun. Pure Appl. Math., 2, 279-301, (1975) · Zbl 0348.60031 · doi:10.1002/cpa.3160280206
[11] Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications, (1998), New York: Springer, New York · Zbl 0896.60013
[12] Hayashi, M.; Watanabe, S., Finite-length analyses for source and channel coding on Markov chains, (2013)
[13] Watanabe, S.; Hayashi, M., Finite-length analysis on tail probability for Markov chain and application to simple hypothesis testing, Ann. Appl. Probab., 27, 811-845, (2017) · Zbl 1368.62235 · doi:10.1214/16-AAP1216
[14] Horssen, M.; Guta, M., Sanov and central limit theorems for output statistics of quantum Markov chains, J. Math. Phys., 56, (2015) · Zbl 1318.81045 · doi:10.1063/1.4907995
[15] Ogata, Y., Large deviations in quantum spin chains, Commun. Math. Phys., 296, 35-68, (2010) · Zbl 1193.82007 · doi:10.1007/s00220-010-0986-y
[16] Budini, A. A.; Turner, R. M.; Garrahan, J. P., Fluctuating observation time ensembles in the thermodynamics of trajectories, J. Stat. Mech., (2014) · doi:10.1088/1742-5468/2014/03/P03012
[17] Guta, M.; Kiukas, J., Equivalence classes and local asymptotic normality in system identification for quantum Markov chains, Commun. Math. Phys., 335, 1397-1428, (2015) · Zbl 1319.81017 · doi:10.1007/s00220-014-2253-0
[18] Schrader, R., Perron–Frobenius theory for positive maps on trace ideals, (Mathematical physics in mathematics and physics) (Siena, 2000) Fields Inst. Commun., 30, 361-378, (2001) · Zbl 1017.47018
[19] Wolf, M. M., Quantum channels & operations: guided tour, (2012)
[20] Yoshida, Y.; Hayashi, M., Mixing and asymptotically decoupling properties in general probabilistic theory, (2018)
[21] Bregman, L., The relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7, 200-217, (1967) · doi:10.1016/0041-5553(67)90040-7
[22] Hayashi, M.; Watanabe, S., Information geometry approach to parameter estimation in Markov chains, Ann. Stat., 44, 1495-1535, (2016) · Zbl 1347.62182 · doi:10.1214/15-AOS1420
[23] Rao, C. R., Linear Statistical Inference and its Applications, (1973), New York: Wiley, New York
[24] Gärtner, J., On large deviations from the invariant measure, Theory Probab. Appl., 22, 24-39, (1977) · Zbl 0375.60033 · doi:10.1137/1122003
[25] Kemeny, J. G.; Snell, J. L., Finite Markov Chains, (1960), New York: Springer, New York · Zbl 0089.13704
[26] Hervé, L.; Ledoux, J.; Patilea, V., A uniform Berry–Esseen theorem on M-estimators for geometrically ergodic Markov chains, Bernoulli, 18, 703-734, (2012) · Zbl 1279.60089 · doi:10.3150/10-BEJ347
[27] Vandergraft, J. S., Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16, 1208-1222, (1968) · Zbl 0186.05701 · doi:10.1137/0116101
[28] Barker, G. P., On matrices having an invariant cone, Czech. Math. J., 22, 49-68, (1972) · Zbl 0311.15011 · doi:10.1016/0024-3795(75)90022-1
[29] Barker, G. P.; Schneider, H., Algebraic Perron–Frobenius theory, Linear Algebr. Appl., 11, 219-233, (1975) · Zbl 0311.15011 · doi:10.1016/0024-3795(75)90022-1
[30] Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A., Generalized no-broadcasting theorem, Phys. Rev. Lett., 99, (2007) · doi:10.1103/PhysRevLett.99.240501
[31] Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A., Teleportation in general probabilistic theories foundations of information flow, Proc. Sympos. Appl. Math., 71, 25-47, (2012) · Zbl 1258.81011
[32] Barrett, J.; Hardy, L.; Kent, A., No signaling and quantum key distribution, Phys. Rev. Lett., 95, (2005) · doi:10.1103/PhysRevLett.95.010503
[33] Gudder, S. P., Stochastic Method in Quantum Mechanics, (1979), Amsterdam: North-Holland, Amsterdam · Zbl 0439.46047
[34] Gudder, S. P., Quantum Probability, (1988), New York: Academic, New York
[35] Kimura, G.; Miyadera, T.; Imai, H., Optimal state discrimination in general probabilistic theories, Phys. Rev. A, 79, (2009) · doi:10.1103/PhysRevA.79.062306
[36] Billingsley, P., Probability and Measure, (1995), New York: Wiley, New York · Zbl 0822.60002
[37] Cappé, O.; Moulines, E.; Ryden, T., Inference in Hidden Markov Models, (2005), New York: Springer, New York · Zbl 1080.62065
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