Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient

TL;DR

This paper analyzes the structure and volume of qubit quantum channels using their Choi representation, introduces an algorithm for uniform channel generation, and investigates the distribution of the quantum Dobrushin coefficient through simulations.

Contribution

It provides explicit volume formulas for qubit channels, an algorithm for uniform sampling, and new insights into the distribution of the quantum Dobrushin coefficient.

Findings

Volume formulas for qubit channels over real and complex spaces

An algorithm for generating uniformly distributed quantum channels

Numerical investigation revealing strange behavior in real state spaces

Abstract

The simplest building blocks for quantum computations are the qbit-qbit quantum channels. In this paper we analyse the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels over real and complex state spaces with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. Moreover an algorithm is presented to generate uniformly distributed channels with respect to the Lebesgue measure, which enables further studies. With this algorithm the distribution of trace-distance contraction coefficient (Dobrushin) is investigated numerically by Monte-Carlo simulations, which leads to some conjectures and points out the strange behaviour of the real state space.