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Proceedings of the Steklov Institute of Mathematics

On the Minimum of the Wehrl Entropy for a Locally Compact Abelian Group

Evgenii Igorevich Zelenov

Published Mar 1, 2024• 2 citations
Publisher

Abstract

A construction of the Wehrl entropy is proposed for an arbitrary locally compact abelian group $$G$$ . It is proved that the Wehrl entropy is not less than a certain nonnegative integer, which is an invariant of the group $$G$$ . The minimum of the Wehrl entropy is attained on coherent states.

DOI
10.1134/s0081543824010097

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A construction of the Wehrl entropy is proposed for an arbitrary locally compact abelian group GGG . It is proved that the Wehrl entropy is not less than a certain nonnegative integer, which is an invariant of the group GGG . The minimum of the Wehrl entropy is attained on coherent states.
A Wehrl entropy construction is proposed for an arbitrary locally compact abelian group GGG. It is proved that the Wehrl entropy is not less than a non-negative integer, which is an invariant of the group GGG. The minimum of the Wehrl entropy is achieved on coherent states.
We focus on quantum systems represented by a Hilbert space L 2 ( A ) , where A is a locally compact Abelian group that contains a compact open subgroup. We examine two interconnected issues related to Weyl-Heisenberg operators. First, we provide a complete and elegant solution to the problem of describing the stabilizer states in terms of their wave functions — an issue that arises in quantum information theory. Subsequently, we demonstrate that the stabilizer states are exactly the minimizers of the Wehrl entropy, thereby solving the Wehrl-type entropy conjecture for any such group (in particular, for finite-dimensional vector spaces over non-Archimedean local fields). Additionally, we construct a moduli space for the set of stabilizer states, that is, a parametrization of this set, that endows it with a natural algebraic structure, and we derive a formula for the number of stabilizer states when A is finite. Indeed, these results are novel even for finite Abelian groups.
Let GGG be a compact connected semisimple Lie group, and let π:G→U(Vλ)π:G\to U(V_λ)π:G→U(Vλ​) be a finite-dimensional irreducible unitary representation associated with the highest weight λλλ. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on VλV_λVλ​. Similar extremal properties hold for power functions in place of the entropy function. The proof uses a second variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the uncertainty principle characterizing coherent states. This solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups.
The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.