# The Wehrl entropy has Gaussian optimizers

## TL;DR

This paper establishes that thermal Gaussian states minimize Wehrl entropy for a given von Neumann entropy, revealing a fundamental link between these entropies and the optimality of Gaussian states in quantum information.

## Contribution

It proves that thermal Gaussian states are the minimizers of Wehrl entropy at fixed von Neumann entropy and characterizes the quantum-classical channel's properties related to Gaussian states.

## Key findings

- Thermal Gaussian states achieve minimum Wehrl entropy for given von Neumann entropy.
- The quantum-classical Husimi Q representation channel is asymptotically equivalent to a Gaussian quantum-limited amplifier.
- The Husimi Q representation of passive states majorizes that of other states with the same spectrum.

## Abstract

We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantum-classical channel that associates to a quantum state its Husimi Q representation is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to determine the p->q norms of the aforementioned quantum-classical channel in the two particular cases of one mode and p=q, and prove that they are achieved by thermal Gaussian states. The same equivalence permits to prove that the Husimi Q representation of a one-mode passive state (i.e. a state diagonal in the Fock basis with eigenvalues decreasing as the energy increases) majorizes the Husimi Q representation of any other one-mode state with the same spectrum, i.e. it maximizes any convex functional.