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arXiv:2306.15357

On the Wehrl entropy lower bound for a locally compact abelian group

Evgenii Igorevich Zelenov

Published Jun 27, 2023• 1 citation
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Abstract

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

DOI
10.48550/arxiv.2306.15357

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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.
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.
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.
We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric SU(N)SU(N)SU(N) coherent states, which was obtained only recently by the authors. Here, we propose an elementary proof of this result, based on reformulating the Wehrl-type entropy as a function defined on the unit sphere in Cd\mathbb{C}^dCd, for some suitable ddd, and on some explicit (and somewhat surprising) computations.