Stability of logarithmic Sobolev inequalities under a noncommutative change of measure

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.

[1]  K. Temme,et al.  Quantum logarithmic Sobolev inequalities and rapid mixing , 2012, 1207.3261.

[2]  B. Pagter,et al.  Double Operator Integrals , 2002 .

[4]  S. Beigi,et al.  Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm , 2015, 1509.02610.

[5]  Michel Ledoux,et al.  Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited , 2001 .

[6]  Marius Junge,et al.  Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups , 2019, IEEE Transactions on Information Theory.

[7]  Daniel Stilck França,et al.  Sandwiched Rényi Convergence for Quantum Evolutions , 2016, 1607.00041.

[8]  Anna Maria Paganoni,et al.  Entropy inequalities for unbounded spin systems , 2002 .

[9]  Li Gao,et al.  Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions , 2018, Annales Henri Poincaré.

[10]  D. Stroock,et al.  Logarithmic Sobolev inequalities and stochastic Ising models , 1987 .

[11]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[12]  Salman Beigi,et al.  Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses , 2018, ArXiv.

[13]  P. Zoller,et al.  Preparation of entangled states by quantum Markov processes , 2008, 0803.1463.

[14]  R. Renner,et al.  Fundamental work cost of quantum processes , 2017, 1709.00506.

[15]  David Pérez-García,et al.  Quantum conditional relative entropy and quasi-factorization of the relative entropy , 2018, Journal of Physics A: Mathematical and Theoretical.

[16]  Michael M. Wolf,et al.  Entropy Production of Doubly Stochastic Quantum Channels , 2015 .

[17]  E. Carlen,et al.  Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems , 2018, Journal of statistical physics.

[18]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[19]  Ivan Bardet,et al.  Estimating the decoherence time using non-commutative Functional Inequalities , 2017, 1710.01039.

[20]  Fernando G. S. L. Brandão,et al.  Quantum Gibbs Samplers: The Commuting Case , 2014, Communications in Mathematical Physics.

[21]  Franco Fagnola,et al.  GENERATORS OF DETAILED BALANCE QUANTUM MARKOV SEMIGROUPS , 2007, 0707.2147.

[22]  Operator integration, perturbations, and commutators , 1993 .

[23]  B. Zegarliński,et al.  Hypercontractivity in Noncommutative LpSpaces , 1999 .

[24]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[25]  H. Spohn.  Entropy production for quantum dynamical semigroups , 1978 .

[26]  I. Stamatescu,et al.  Decoherence and the Appearance of a Classical World in Quantum Theory , 1996 .

[27]  L. Miclo,et al.  Remarques sur l’hypercontractivité et l’évolution de l’entropie pour des chaînes de Markov finies , 1997 .

[28]  V. Umanità,et al.  Decoherence for Quantum Markov Semi-Groups on Matrix Algebras , 2013 .

[29]  Sergey G. Bobkov,et al.  Modified log-sobolev inequalities, mixing and hypercontractivity , 2003, STOC '03.

[30]  K. Temme.  Thermalization Time Bounds for Pauli Stabilizer Hamiltonians , 2014, 1412.2858.

[31]  F. Pastawski,et al.  Hypercontractivity of quasi-free quantum semigroups , 2014, 1403.5224.

[32]  Daniel Stilck França,et al.  Relative Entropy Convergence for Depolarizing Channels , 2015, 1508.07021.

[33]  D. Potapov,et al.  Double Operator Integrals and Submajorization , 2010 .

[34]  Maxim Raginsky,et al.  Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels , 2014, IEEE Transactions on Information Theory.

[36]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[37]  Ivan Bardet,et al.  Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates , 2018, Annales Henri Poincaré.

[38]  G. Lindblad.  Expectations and entropy inequalities for finite quantum systems , 1974 .

[39]  F. Hiai,et al.  From quasi-entropy to various quantum information quantities , 2012 .

[40]  E. Carlen,et al.  Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance , 2016, 1609.01254.

[41]  N. Nikolski,et al.  LIPSCHITZ AND COMMUTATOR ESTIMATES IN SYMMETRIC OPERATOR SPACES. , 2008 .

[42]  Maurizio Verri,et al.  Long-time asymptotic properties of dynamical semigroups onW*-algebras , 1982 .