Balanced distribution-energy inequalities and related entropy bounds

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 ${\mathbb{R}}^n$. 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 ${\mathbb{R}}^n$, 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 $A$.

21 pages

References in corpus (1)

• Spectral density and Sobolev inequalities for pure and mixed states

Cited by in corpus (24)

• Entropic Uncertainty Relations and their Applications
• Position-Momentum Uncertainty Relations in the Presence of Quantum Memory
• A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics
• Wehrl entropy, entropic uncertainty relations and entanglement
• Hardy and Lieb-Thirring inequalities for anyons
• Entropy and the uncertainty principle
• Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics
• A positive density analogue of the Lieb-Thirring inequality
• Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems
• Universal Functionals in Density Functional Theory
• Hartree Corrections in a Mean-field Limit for Fermions with Coulomb Interaction
• Lieb-Thirring inequality with semiclassical constant and gradient error term
• The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities
• Relative entropic uncertainty relation
• Relative entropic uncertainty relation for scalar quantum fields
• An entropic uncertainty principle for positive operator valued measures
• Methods of modern mathematical physics: Uncertainty and exclusion principles in quantum mechanics
• Energy contribution of a point interacting impurity in a Fermi gas
• A simple approach to Lieb--Thirring type inequalities
• A proof of the Lieb-Thirring inequality via the Besicovitch covering lemma
• Global existence versus finite time blowup dichotomy for the system of nonlinear Schrödinger equations
• Quantum Uncertainty and Entropy
• On the number and sums of eigenvalues of Schrödinger-type operators with degenerate kinetic energy
• Book review: Stability of Matter in Quantum Mechanics, by Elliott H. Lieb and Robert Seiringer