We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch–Gordan theory and the reciprocity theorem associated with induced unitary group representations. Applications to several examples involving SU(2), SU(3), and the Heisenberg–Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase–space description of quantum mechanics and quantum state reconstruction problems.We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch–Gordan theory and the reciprocity theorem associated with induced unitary group representations. Applications to several examples involving SU(2), SU(3), and the Heisenberg–Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase–space description of quantum mechanics and quantum state reconstruction problems.

arXiv:quant-ph/0002070Journal of Mathematical PhysicsMay 21, 20038 citations

Coherent States for Arbitrary Lie Groups

A. M. Perelomov

Springer eBooksJan 1, 198625 citations

Wehrl-type entropy problem for compact connected semisimple Lie groups

Haonan Zhang

Let $G$ be a compact connected semisimple Lie group, and let $π:G\to U(V_λ)$ be a finite-dimensional irreducible unitary representation associated with the highest weight $λ$ . We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_λ$ . 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.Let $G$ be a compact connected semisimple Lie group, and let $π:G\to U(V_λ)$ be a finite-dimensional irreducible unitary representation associated with the highest weight $λ$. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_λ$. 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.

arXiv:2609.18919Sep 16, 20260 citations

SU(2) and SU(1,1) group parametrizations via qubit induced correlated probabilities

Alexey A. Strakhov, Vladimir I. Man’ko

Abstract A group is an algebraic structure equipped with binary associative multiplication, identity element and inverse operation with respect to identity. Lie group is a smooth manifold with group multiplication and the inverse being smooth functions. Therefore as one chooses manifold coordinate charts, various parametrizations of multiplication and inverse operations arise. Recently, the simple probability representation of qubit states and observables and an associated triangle geometrical picture called ‘quantum suprematism’ were introduced. At its core lies the representation of the Bloch sphere by correlated dichotomic probability distribution. The aim of the present paper is the introduction of probability parametrization of two classical compact and non-compact Lie groups, SU(2) and SU(1,1) respectively.Abstract A group is an algebraic structure equipped with binary associative multiplication, identity element and inverse operation with respect to identity. Lie group is a smooth manifold with group multiplication and the inverse being smooth functions. Therefore as one chooses manifold coordinate charts, various parametrizations of multiplication and inverse operations arise. Recently, the simple probability representation of qubit states and observables and an associated triangle geometrical picture called ‘quantum suprematism’ were introduced. At its core lies the representation of the Bloch sphere by correlated dichotomic probability distribution. The aim of the present paper is the introduction of probability parametrization of two classical compact and non-compact Lie groups, SU(2) and SU(1,1) respectively.

Journal of Physics A Mathematical and TheoreticalSep 10, 20192 citations

Let

G

be a compact connected semisimple Lie group, and let

π:G\to U(V_λ)

be a finite-dimensional irreducible unitary representation associated with the highest weight

λ

. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on

V_λ

. 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.