Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

A short quantum Markov chain is a tripartite state $\rho_{ABC}$ such that system $A$ can be recovered perfectly by acting on system $C$ of the reduced state $\rho_{BC}$. Such states have conditional mutual information $I(A;B|C)$ equal to zero and are the only states with this property. A quantum channel $\mathcal{N}$ is sufficient for two states $\rho$ and $\sigma$ if there exists a recovery channel using which one can perfectly recover $\rho$ from $\mathcal{N}(\rho)$ and $\sigma$ from $\mathcal{N}(\sigma)$. The relative entropy difference $D(\rho\Vert\sigma)-D(\mathcal{N}(\rho)\Vert\mathcal{N}(\sigma))$ is equal to zero if and only if $\mathcal{N}$ is sufficient for $\rho$ and $\sigma$. In this paper, we show that these properties extend to Renyi generalizations of these information measures which were proposed in [Berta et al., J. Math. Phys. 56, 022205, (2015)] and [Seshadreesan et al., J. Phys. A 48, 395303, (2015)], thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Renyi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.Comment: v4: 26 pages, 1 figure; reorganized and one open question solved with Choi's inequality (at the suggestion of an anonymous referee

Gibbs conditioning extended, Boltzmann conditioning introduced

Conditional Equi-concentration of Types on I-projections (ICET) and Extended Gibbs Conditioning Principle (EGCP) provide an extension of Conditioned Weak Law of Large Numbers and of Gibbs Conditioning Principle to the case of non-unique Relative Entropy Maximizing (REM) distribution (aka I-projection). ICET and EGCP give a probabilistic justification to REM under rather general conditions. mu-projection variants of the results are introduced. They provide a probabilistic justification to Maximum Probability (MaxProb) method. 'REM/MaxEnt or MaxProb?' question is discussed, briefly. Jeffreys Conditioning Principle is mentioned.Comment: Three major changes: 1) Definition of proper I-projection has been changed. 2) An argument preceding Eq. (7) at the proof of ICET is now correctly stated. 3) Abstract was rewritten. To appear at Proceedings of MaxEnt 2004 worksho

Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $\xi(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $\xi$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $\xi(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $\xi(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics