This paper extends the Bakry-& Eacute,.mery criterion relating the Ricci curvature and logarithmic Sobolev inequalities to the noncommutative setting. We obtain easily computable complete modified logarithmic Sobolev inequalities of graph Laplacians and Lindblad operators of the corresponding graph H & ouml,.rmander systems. We develop the anti-transference principle stating that the matrix-valued modified logarithmic Sobolev inequalities of sub-Laplacian operators on a compact Lie group are equivalent to such inequalities of a family of the transferred Lindblad operators with a uniform lower bound.

Hirche and Tomamichel recently introduced quantum f-divergence as an integral of quantum Hockey stick divergence. In this paper, we study the optimization of quantum f-divergence between the unitary orbits. The proof relies on the well-known Lidskii's inequality. We also generalize the result to the mixed unitary orbits.

We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a H & ouml,.rmander system on a compact manifold satisfies a modified log-Sobolev inequality uniformly for scalar and matrix-valued functions. For quantum Markov semigroups, we show that the complete modified logarithmic Sobolev constant is comparable to the spectral gap up to the logarithm of the dimension. Such estimates are asymptotically tight for a quantum birth-death process. Our results, along with the consequence of concentration inequalities, are applicable to GNS-symmetric semigroups on general von Neumann algebras.

We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output by some quantum operation, if and only if, the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger-Yanase-Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analogs of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying monotone metrics on the quantum state space, i.e. Riemannian metrics non-increasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum chi 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi <^>2$$\end{document} divergence. We obtain an approximate recovery result in the sense that, if the quantum chi 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi <^>2$$\end{document} divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the chi 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\frac{1}{2}}$$\end{document} divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.