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 from its distribution in the partition and the spectral density of A. These bounds improve with the renement of the partition, and generalize inequalities by Li-Yau and Lieb-Thirring for the Laplacian in Rn . They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of , as measured from X, and some spectral entropy, with respect to its energy distribution. On Rn , 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 A.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Michel Rumin
• Keywords: CORR 2010, Education, Lower Bounds, Mixed States, Self-adjoint Operator