Dynamical Entropy of Quasi-local Algebras

Philippe Blanchard, Mario Hellmich, Piotr Ługiewicz +1 more

Ángela Capel, Angelo Lucia, David Pérez-García

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- σ -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.

Stefan Teufel, Marius Wesle, Tom Wessel

We prove global existence and uniqueness of dynamics on the quasi-local algebra $\mathcal{A}$ of a quantum lattice system for spatially growing derivations $\mathcal{L}_Φ= \sum_x [ Φ_x , \cdot ]$. Existing results assume that the local terms $Φ_x\in\mathcal{A}$ of the generator are uniformly bounded in space with respect to appropriate weighted norms $\lVert Φ_x \rVert_{G,x}$. Analogous to the global existence result for first order ODEs, we show that global existence and uniqueness persist if the size of the local terms $\lVert Φ_x \rVert_{G,x}$ grows at most linearly in space. This considerably enlarges the class of derivations known to have well-defined dynamics. Moreover, we obtain Lieb-Robinson bounds with exponential light cones for such dynamics. For the proof, we assume Lieb-Robinson bounds with linear light cones for dynamics, whose generators have uniformly bounded local terms. Such bounds are known to hold, for example, if the local terms are of finite range or exponentially localized.

Masanori Ohya, Dėnes Petz

Francesco Fidaleo

R. F. Streater

Gives a short proof that any infinitely divisible even representation of a Clifford algebra or a CAR algebra is quasi-free, and of Hudson's central limit theorem (1973) for fermions. The author shows that the entropy of any even state of the CAR, Clifford or CCR algebra is less than or equal to the entropy of the quasi-free state with the same two-point function.