Entropy Densities for Gibbs States and Algebraic States : A joint work with D.Petz

Hajime Moriya

Li Gao, Jingyu Guo, Sihao Wang

We prove a complete modified logarithmic Sobolev inequality with a system-size-independent constant for Chen--Kastoryano--Gilyén Gibbs samplers on one-dimensional spin chain at every fixed positive temperature. The result holds for Hamiltonians with bounded finite-range interactions, including noncommuting and spatially nonuniform interactions. It yields trace and diamond mixing times of order $O(\log N+\log(1/ε))$. The proof establishes complete spatial entropy factorization and absorbs quasi-local errors as conditional relative entropy at the original joint state and reference. The static estimate also yields a system-size-independent CMLSI for the single-site Petz heat-bath and regularised heat-bath dynamics.

George Androulakis, Tiju Cherian John

Fumio Hiai, D�nes Petz

V. P. Belavkin, P. Staszewski

Teruaki Nagasawa, Kohtaro Kato, Eyuri Wakakuwa +1 more

Observational entropy—a quantity that unifies Boltzmann's entropy, Gibbs' entropy, von Neumann's macroscopic entropy, and the diagonal entropy—was recently argued to play a key role in a modern formulation of statistical mechanics. Here, relying on algebraic techniques taken from Petz's theory of statistical sufficiency and on a Lévy-type concentration bound, we prove rigorous theorems showing how the observational entropy of a system undergoing a unitary evolution chosen at random tends to increase with overwhelming probability and to reach its maximum very quickly. More precisely, we show that for any observation that is sufficiently coarse with respect to the size of the system, regardless of the initial state of the system (be it pure or mixed), random evolution renders its state practically indistinguishable from the uniform (i.e., maximally mixed) distribution with a probability approaching 1 as the size of the system grows. The same conclusion holds not only for random evolutions sampled according to the unitarily invariant Haar distribution but also for approximate 2-designs, which are thought to provide a more physically and computationally reasonable model of random evolutions. Published by the American Physical Society 2024