Balanced distribution-energy inequalities and related entropy bounds

TL;DR

This paper establishes new lower bounds on the energy of mixed states based on their distribution and spectral density, generalizing classical inequalities and deriving an uncertainty principle and entropy bounds without positivity assumptions.

Contribution

It introduces generalized energy and entropy inequalities for self-adjoint operators, extending classical results like Li-Yau and Lieb--Thirring to broader contexts.

Findings

Improved lower bounds on energy using partition refinement.

Derived an uncertainty principle linking spatial and spectral entropy.

Established a general log-Sobolev inequality for mixed states.

Abstract

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 $\rho$ 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 $\R^n$. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of $\rho$, as seen from $X$, and some spectral entropy, with respect to its energy distribution. On $\R^n$, this yields lower bounds on the sum of the entropy of the densities of $\rho$ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on $A$.