1Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France
2Department of Mathematics, Technische Universität München, 85748 Garching, Germany
3Munich Center for Quantum Science and Technology (MCQST), München, Germany
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Abstract
We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.
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[1] É. Amorim and E. A. Carlen. Complete positivity and self-adjointness. Linear Algebra and its Applications, 611:389–439, 2021.
https://doi.org/10.1016/j.laa.2020.10.038
[2] Ángela Capel, C. Rouzé, and D. S. França. The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions, 2021.
arXiv:2009.11817
[3] S. Attal and Y. Pautrat. From Repeated to Continuous Quantum Interactions. Annales Henri Poincaré, 7:59–104, Jan. 2006.
https://doi.org/10.1007/s00023-005-0242-8
[4] A. Barchielli and A. Holevo. Constructing quantum measurement processes via classical stochastic calculus. Stochastic Processes and their Applications, 58(2):293–317, Aug. 1995.
https://doi.org/10.1016/0304-4149(95)00011-U
[5] I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-Garcia, and C. Rouzé. Entropy decay for Davies semigroups of a one dimensional quantum lattice. in preparation, 2021.
https://doi.org/10.48550/arXiv.2112.00601
[6] I. Bardet, Á. Capel, A. Lucia, D. Pérez-Garcia, and C. Rouzé. On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems. Journal of Mathematical Physics, 62(6):061901, June 2021.
https://doi.org/10.1063/1.5142186
[7] I. Bardet, Á. Capel, and C. Rouzé. Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations. Annales Henri Poincaré, July 2021.
https://doi.org/10.1007/s00023-021-01088-3
[8] I. Bardet and C. Rouzé. Hypercontractivity and logarithmic Sobolev inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates. In Annales Henri Poincaré, pages 1–65. Springer, 2022.
https://doi.org/10.1007/s00023-022-01196-8
[9] S. Beigi, N. Datta, and C. Rouzé. Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses. Communications in Mathematical Physics, 376(2):753–794, May 2020.
https://doi.org/10.1007/s00220-020-03750-z
[10] T. Benoist, N. Cuneo, V. Jakšić, Y. Pautrat, and C.-A. Pillet. On the nature of the quantum detailed balance condition. In preparation.
[11] I. Bjelaković, J.-D. Deuschel, T. Krüger, R. Seiler, R. Siegmund-Schultze, and A. Szkoła. A quantum version of Sanov’s theorem. Communications in mathematical physics, 260(3):659–671, 2005.
https://doi.org/10.1007/s00220-005-1426-2
[12] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. Journal of Functional Analysis, 163(1):1–28, 1999.
https://doi.org/10.1016/0304-4149(95)00011-u/10.1006/jfan.1998.3326
[13] L. Bouten, R. V. Handel, and M. R. James. An Introduction to Quantum Filtering. SIAM Journal on Control and Optimization, 46(6):2199–2241, Jan. 2007.
https://doi.org/10.1137/060651239
[14] D. Burgarth, G. Chiribella, V. Giovannetti, P. Perinotti, and K. Yuasa. Ergodic and mixing quantum channels in finite dimensions. New Journal of Physics, 15(7):073045, jul 2013.
https://doi.org/10.1088/1367-2630/15/7/073045
[15] R. Carbone and A. Martinelli. Logarithmic Sobolev inequalities in non-commutative algebras. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18(02):1550011, 2015.
https://doi.org/10.1142/S0219025715500113
[16] E. A. Carlen and J. Maas. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis, 273(5):1810–1869, Sept. 2017.
https://doi.org/10.1016/j.jfa.2017.05.003
[17] E. A. Carlen and J. Maas. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. Journal of Statistical Physics, 178(2):319–378, 2020.
https://doi.org/10.1007/s10955-019-02434-w
[18] J. Dalibard, Y. Castin, and K. Mølmer. Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett., 68(5):580, Feb. 1992.
https://doi.org/10.1103/PhysRevLett.68.580
[19] N. Datta and C. Rouzé. Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality. Annales Henri Poincaré, 21(7):2115–2150, Feb. 2020.
https://doi.org/10.1007/s00023-020-00891-8
[20] E. B. Davies. One-parameter semigroups. Academic Press, London New York, 1980.
[21] G. De Palma, M. Marvian, D. Trevisan, and S. Lloyd. The quantum Wasserstein distance of order 1. IEEE Transactions on Information Theory, 67(10):6627–6643, 2021.
https://doi.org/10.1109/TIT.2021.3076442
[22] G. De Palma and C. Rouzé. Quantum concentration inequalities. In Annales Henri Poincaré, pages 1–39. Springer, 2022.
https://doi.org/10.1007/s00023-022-01181-1
[23] G. De Palma and D. Trevisan. Quantum optimal transport with quantum channels. In Annales Henri Poincaré, volume 22, pages 3199–3234. Springer, 2021.
https://doi.org/10.1007/s00023-021-01042-3
[24] F. Den Hollander. Large deviations, volume 14. American Mathematical Soc., 2008.
[25] J. Dereziński and W. De Roeck. Extended Weak Coupling Limit for Pauli-Fierz Operators. Communications in Mathematical Physics, 279(1):1–30, Apr. 2008.
https://doi.org/10.1103/10.1007/s00220-008-0419-3
[26] J.-D. Deuschel and D. W. Stroock. Large deviations, volume 342. American Mathematical Soc., 2001.
[27] M. D. Donsker and S. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I. Communications on Pure and Applied Mathematics, 28(1):1–47, 1975.
https://doi.org/10.1002/cpa.3160280102
[28] F. Fagnola and V. Umanità. Generators of detailed balance quantum Markov semigroups. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 10(03):335–363, 2007.
https://doi.org/10.1142/S0219025707002762
[29] F. Fagnola and V. Umanità. Generators of KMS Symmetric Markov Semigroups on $B(mathrm h)$ Symmetry and Quantum Detailed Balance. Communications in Mathematical Physics, 298(2):523–547, 2010.
https://doi.org/10.1007/s00220-010-1011-1
[30] M. Fathi and Y. Shu. Curvature and transport inequalities for Markov chains in discrete spaces. Bernoulli, 24(1), Feb. 2018.
https://doi.org/10.3150/16-bej892
[31] L. Gao, M. Junge, and N. LaRacuente. Fisher information and logarithmic Sobolev inequality for matrix-valued functions. In Annales Henri Poincaré, volume 21, pages 3409–3478. Springer, 2020.
https://doi.org/10.1007/s00023-020-00947-9
[32] L. Gao and C. Rouzé. Ricci curvature of quantum channels on non-commutative transportation metric spaces. arXiv preprint arXiv:2108.10609, 2021.
https://doi.org/10.48550/arXiv.2108.10609
arXiv:2108.10609
[33] L. Gao and C. Rouzé. Complete entropic inequalities for quantum Markov chains. Archive for Rational Mechanics and Analysis, pages 1–56, 2022.
https://doi.org/10.1007/s00205-022-01785-1
[34] N. Gisin and I. C. Percival. The quantum-state diffusion model applied to open systems. Journal of Physics A: Mathematical and General, 25(21):5677–5691, nov 1992.
https://doi.org/10.1088/0305-4470/25/21/023
[35] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. Journal of Mathematical Physics, 17(5):821–825, 1976.
https://doi.org/10.1063/1.522979
[36] N. Gozlan and C. Léonard. A large deviation approach to some transportation cost inequalities. Probability Theory and Related Fields, 139(1):235–283, Sep 2007.
https://doi.org/10.1007/s00440-006-0045-y
[37] A. Guillin, C. Léonard, L. Wu, and N. Yao. Transportation-information inequalities for Markov processes. Probability Theory and Related Fields, 144(3):669–695, Jul 2009.
https://doi.org/10.1007/s00440-008-0159-5
[38] E. P. Hanson, C. Rouzé, and D. S. França. Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times. Annales Henri Poincaré, 21(5):1517–1571, Mar. 2020.
https://doi.org/10.1007/s00023-020-00906-4
[39] A. S. Holevo. Statistical Structure of Quantum Theory. Springer Berlin Heidelberg, 2001.
https://doi.org/10.1007/3-540-44998-1
[40] R. L. Hudson and K. R. Parthasarathy. Quantum Ito’s formula and stochastic evolutions. Communications in mathematical physics, 93(3):301–323, 1984.
https://doi.org/10.1007/BF01258530
[41] R. L. Hudson and K. R. Parthasarathy. Stochastic dilations of uniformly continuous completely positive semigroups. In Positive Semigroups of Operators, and Applications, pages 353–378. Springer, 1984.
https://doi.org/10.1007/BF02280859
[42] V. Jakšić, C.-A. Pillet, and M. Westrich. Entropic fluctuations of quantum dynamical semigroups. J. Stat. Phys., 154(1-2):153–187, 2014.
https://doi.org/10.1007/s10955-013-0826-5
[43] M. Junge and Q. Zeng. Noncommutative martingale deviation and Poincaré type inequalities with applications. Probability Theory and Related Fields, 161(3-4):449–507, 2015.
https://doi.org/10.1007/s00440-014-0552-1
[44] M. J. Kastoryano and F. G. S. L. Brandão. Quantum Gibbs Samplers: The Commuting Case. Communications in Mathematical Physics, 344(3):915–957, 2016.
https://doi.org/10.1007/s00220-016-2641-8
[45] M. J. Kastoryano and K. Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. Journal of Mathematical Physics, 54(5), 2013.
https://doi.org/10.1063/1.4804995
[46] C. King. Hypercontractivity for Semigroups of Unital Qubit Channels. Communications in Mathematical Physics, 328(1):285–301, Mar. 2014.
https://doi.org/10.1007/s00220-014-1982-4
[47] B. Kümmerer and H. Maassen. A pathwise ergodic theorem for quantum trajectories. Journal of Physics A: Mathematical and General, 37(49):11889–11896, nov 2004.
https://doi.org/10.1088/0305-4470/37/49/008
[48] D. Levin and Y. Peres. Markov Chains and Mixing Times. American Mathematical Society, Oct. 2017.
https://doi.org/10.1090/mbk/107
[49] G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2):119–130, 1976.
https://doi.org/10.1007/BF01608499
[50] E. Lukacs and K. M. R. Collection. Characteristic Functions. Griffin books of cognate interest. Griffin, 1970.
[51] K. Marton. A simple proof of the blowing-up lemma. IEEE Transactions on Information Theory, 32(3):445–446, May 1986.
https://doi.org/10.1109/TIT.1986.1057176
[52] A. Müller-Hermes, D. S. França, and M. M. Wolf. Relative entropy convergence for depolarizing channels. Journal of Mathematical Physics, 57(2):022202, Feb. 2016.
https://doi.org/10.1063/1.4939560
[53] R. Olkiewicz and B. Zegarlinski. Hypercontractivity in Noncommutative Lp Spaces. Journal of Functional Analysis, 161(1):246–285, 1999.
https://doi.org/10.1006/jfan.1998.3342
[54] Y. Ollivier. Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3):810–864, Feb. 2009.
https://doi.org/10.1016/j.jfa.2008.11.001
[55] G. D. Palma and S. Huber. The conditional entropy power inequality for quantum additive noise channels. Journal of Mathematical Physics, 59(12):122201, Dec. 2018.
https://doi.org/10.1063/1.5027495
[56] K. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Springer Basel, 1992.
https://doi.org/10.1007/978-3-0348-0566-7
[57] C. Rouzé and N. Datta. Concentration of quantum states from quantum functional and transportation cost inequalities. Journal of Mathematical Physics, 60(1):012202, 2019.
https://doi.org/10.1063/1.5023210
[58] K. Temme, F. Pastawski, and M. J. Kastoryano. Hypercontractivity of quasi-free quantum semigroups. Journal of Physics A: Mathematical and Theoretical, 47(40):405303, Sept. 2014.
https://doi.org/10.1088/1751-8113/47/40/405303
[59] M. van Horssen and M. Guţă. Sanov and central limit theorems for output statistics of quantum Markov chains. Journal of Mathematical Physics, 56(2):022109, Feb. 2015.
https://doi.org/10.1063/1.4907995
[60] C. Villani. Topics in optimal transportation. Number 58. American Mathematical Soc., 2003.
[61] H. M. Wiseman and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2009.
https://doi.org/10.1017/CBO9780511813948
[62] M. Wolf. Quantum channels & operations: Guided tour. Lecture notes available at http://www-m5. ma. tum. …, 2011.
https://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf
[63] L. Wu. Feynman-Kac Semigroups, Ground State Diffusions, and Large Deviations. Journal of Functional Analysis, 123(1):202–231, July 1994.
https://doi.org/10.1006/jfan.1994.1087
[64] L. Wu. A deviation inequality for non-reversible Markov processes. Annales de l’I.H.P. Probabilités et statistiques, 36(4):435–445, 2000.
https://doi.org/10.1016/S0246-0203(00)00135-7
Cited by
[1] Bowen Li and Jianfeng Lu, “Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics”, arXiv:2207.06422.
[2] Federico Girotti, Juan P. Garrahan, and Mădălin Guţă, “Concentration Inequalities for Output Statistics of Quantum Markov Processes”, arXiv:2206.14223.
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