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arXiv:nlin/0208007 · Journal of Physics A Mathematical and General

Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one

Ayumu Sugita

Published Oct 8, 2002• 20 citations
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Abstract

Gnutzmann and Życzkowski have proposed the Rényi–Wehrl entropy as a generalization of the Wehrl entropy, and conjectured that its minimum is obtained for coherent states. We prove this conjecture for the Rényi index q = 2, 3, ... in the cases of compact semisimple Lie groups. A general formula for the minimum value is given.
DOI
10.1088/0305-4470/35/42/105

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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.
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.
Abstract. We generalize the concept of the Wehrl entropy of quantum states which gives a basis–independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi–Wehrl entropies for pure states for spin systems. According to Lieb’s conjecture the minimal values are provided by the spin coherent states. Though Lieb’s conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi–Wehrl entropies averaged over the natural measure in the space of pure quantum states. 1.
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.
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.
We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.