The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere

TL;DR

This paper investigates the possible ranges of combined Hilbert-Schmidt norms of quantum channels and their complements, characterizes extremal channels, and explores matrix integrals over the unit sphere with applications to infinite-dimensional systems.

Contribution

It provides a detailed analysis of the norm ranges of quantum channels and their complements, characterizes channels achieving extremal values, and establishes equivalences of matrix integrals over the sphere.

Findings

Characterization of quantum channels with extremal Hilbert-Schmidt norm sums.

Description of pure state preserving maps in infinite dimensions.

Equivalence results for matrix integrals over the unit sphere.

Abstract

The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of $\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2$ for quantum channels $\mathcal{E}$ and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where $\|\mathcal{E}\|_2$ is the Hilbert-Schmidt norm of $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ is a complementary channel of $\mathcal{E}.$ Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained.