Quantum conditional relative entropy and quasi-factorization of the relative entropy

Abstract

The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$\sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.

arXiv commentv2: Final version, minor changes, 39 pages, 2 figures

References in corpus (18)

• Preparation of Entangled States by Quantum Markov Processes
• On quantum Renyi entropies: a new generalization and some properties
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Quantum conditional mutual information and approximate Markov chains
• Monotonicity of quantum relative entropy revisited
• alpha-z-relative Renyi entropies
• Quantum logarithmic Sobolev inequalities and rapid mixing
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• Universal recovery maps and approximate sufficiency of quantum relative entropy
• Multivariate Trace Inequalities
• Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution
• Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps
• Hypercontractivity of quasi-free quantum semigroups
• Fundamental work cost of quantum processes
• Relative Entropy Convergence for Depolarizing Channels
• Entropy Production of Doubly Stochastic Quantum Channels
• Superadditivity of quantum relative entropy for general states
• Swiveled Rényi entropies

Cited by in corpus (16)

• Universal Sampling Lower Bounds for Quantum Error Mitigation
• Rapid thermalization of spin chain commuting Hamiltonians
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems
• Continuity of quantum entropic quantities via almost convexity
• Approximate tensorization of the relative entropy for noncommuting conditional expectations
• Stability of logarithmic Sobolev inequalities under a noncommutative change of measure
• A strengthened data processing inequality for the Belavkin-Staszewski relative entropy
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Quantum computational advantage with constant-temperature Gibbs sampling
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Weak quasi-factorization for the Belavkin-Staszewski relative entropy
• Strong decay of correlations for Gibbs states in any dimension
• Lower semicontinuity of the relative entropy disturbance and its corollaries
• New monotonicity property of the quantum relative entropy
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains