1 December 2011 Balanced distribution-energy inequalities and related entropy bounds
Michel Rumin
Duke Math. J. 160(3): 567-597 (1 December 2011). DOI: 10.1215/00127094-1444305


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 and Yau and by Lieb and Thirring for the Laplacian in Rn. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, 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 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.


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Michel Rumin. "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


Published: 1 December 2011
First available in Project Euclid: 7 November 2011

zbMATH: 1239.47019
MathSciNet: MR2852369
Digital Object Identifier: 10.1215/00127094-1444305

Primary: 46E35 , 47B06 , 58J50
Secondary: 35P20 , 94A17

Rights: Copyright © 2011 Duke University Press


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Vol.160 • No. 3 • 1 December 2011
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