Balanced distribution-energy inequalities and related entropy bounds

Let A be a self-adjoint operator acting over a space X endowed with a partition. We give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. These bounds improve with the refinement of the partition, and generalize inequalities by Li-Yau and Lieb–Thirring for the Laplacian in R. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, as seen from X, and some spectral entropy, with respect to its energy distribution. On R, this yields lower bounds on the sum of the entropy of the densities of ρ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on A.