Entropy and the uncertainty principle

TL;DR

This paper generalizes and unifies theorems related to classical entropies of quantum states in different bases, providing sharp uncertainty principles that are exact in the semi-classical limit.

Contribution

It extends and refines existing entropy uncertainty theorems, offering a unified framework with sharp bounds applicable in high-temperature regimes.

Findings

Theorems are generalized and improved.

Results are sharp in the semi-classical limit.

Provides a unified approach to entropy-based uncertainty principles.

Abstract

We generalize, improve and unify theorems of Rumin, and Maassen--Uffink about classical entropies associated to quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different bases. Thus they provide a kind of uncertainty principle. Our inequalities are sharp because they are exact in the high-temperature or semi-classical limit.