Entropy of semiclassical measures of the Walsh-quantized baker's map

TL;DR

This paper investigates the localization and entropy properties of eigenstates in the Walsh-quantized baker's map, revealing limitations of quantum ergodicity and establishing entropy bounds using an entropic uncertainty principle.

Contribution

It provides new lower bounds on the entropy of semiclassical measures for the Walsh-quantized baker's map, highlighting non-ergodic behavior in this quantum chaotic model.

Findings

Quantum unique ergodicity fails for the Walsh baker's map.

Lower bounds on semiclassical measure entropies are established.

The entropic uncertainty principle is used as a key analytical tool.

Abstract

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical regime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".