Entropy of quantum limits for symplectic linear maps of the multidimensional torus

TL;DR

This paper establishes a lower bound on the entropy of semiclassical measures for quantizable symplectic linear maps on multidimensional tori, linking quantum limits to classical dynamical properties.

Contribution

It provides an explicit entropy lower bound for semiclassical measures associated with quantizable symplectic matrices, revealing constraints on quantum states based on classical eigenvalues.

Findings

Lower bound on entropy for semiclassical measures

Semiclassical measures cannot concentrate on closed orbits if eigenvalues lie outside the unit circle

Quantum limits are constrained by classical spectral properties

Abstract

In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A.