Functional calculus of quantum channels for the holomorphic discrete series of $SU(1,1)$

TL;DR

This paper develops a functional calculus for quantum channels associated with the holomorphic discrete series of $SU(1,1)$, using limit formulas involving Husimi functions and Berezin transforms to analyze tensor product decompositions.

Contribution

It introduces equivariant quantum channels for each component of the tensor product of holomorphic discrete series representations of $SU(1,1)$ and establishes a limit formula for their functional calculus.

Findings

Derived a limit formula for the trace of the functional calculus.

Expressed the limit using generalized Husimi functions.

Connected the limit to Berezin transforms.

Abstract

The tensor product of two holomorphic discrete series representations of $SU(1,1)$ can be decomposed as a direct sum of infinitely many discrete series. I shall introduce equivariant quantum channels for each component of the direct sum, mapping bounded operators on one factor of the tensor product to operators on the component. Next I prove a limit formula for the trace of the functional calculus and I prove that the limit can be expressed using generalized Husimi functions or using Berezin transforms.