Let be a self-adjoint operator acting over a space endowed with a partition. We give lower bounds on the energy of a mixed state from its distribution in the partition and the spectral density of . These bounds improve with the refinement of the partition, and generalize inequalities by Li and Yau and by Lieb and Thirring for the Laplacian in . They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of , as measured from , and some spectral entropy, with respect to its energy distribution. On , this yields lower bounds on the sum of the entropy of the densities of and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on .
"Balanced distribution-energy inequalities and related entropy bounds." Duke Math. J. 160 (3) 567 - 597, 1 December 2011. https://doi.org/10.1215/00127094-1444305