Complete Modified Logarithmic Sobolev Inequalities for one-dimensional quantum Spin Chain

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 $O(\log N+\log (1\mathrm{/}\epsilon ))$. 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.