Geometric inequalities from phase space translations

TL;DR

This paper develops quantum analogs of classical geometric inequalities, linking Fisher information and entropy power for quantum states, and applies these results to quantum heat diffusion and convergence analysis.

Contribution

It introduces a quantum Fisher information inequality based on phase space convolutions, extending classical inequalities to the quantum setting.

Findings

Proves a quantum isoperimetric inequality relating Fisher information and entropy power.

Derives an entropy power inequality for a quantum phase space convolution.

Establishes a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup.

Abstract

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states.