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Limit formulas for the trace of the functional calculus of quantum channels for $SU(2)$

Robin van Haastrecht

Abstract

Lieb and Solovej \cite{liebsolBloch} studied traces of quantum channels, defined by the leading component in the decomposition of the tensor product of two irreducible representations of $SU(2)$, to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. It is proved that the integral is the limit of the trace of the functional calculus of quantum channels. In this paper, we introduce new quantum channels for all the components in the tensor product and generalize their limit formula. We prove that the limit can be expressed using Berezin transforms.

Limit formulas for the trace of the functional calculus of quantum channels for $SU(2)$

Abstract

Lieb and Solovej \cite{liebsolBloch} studied traces of quantum channels, defined by the leading component in the decomposition of the tensor product of two irreducible representations of , to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. It is proved that the integral is the limit of the trace of the functional calculus of quantum channels. In this paper, we introduce new quantum channels for all the components in the tensor product and generalize their limit formula. We prove that the limit can be expressed using Berezin transforms.
Paper Structure (4 sections, 15 theorems, 144 equations)

This paper contains 4 sections, 15 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum channels & Berezin transforms
  4. Functional calculus of quantum channels

Key Result

Theorem 1.1

Let $\phi \in C([0,1])$. Then for any $f \in C(\mathbb{CP}^1)$ such that $\int_{\mathbb{CP}^1} f(z) dz = 1$ and with Toeplitz operator $R_{\mu}^*(f) \geq 0$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 31 more