Volume of the space of qubit-qubit channels and state transformations under random quantum channels

TL;DR

This paper investigates the structure and volume of qubit-qubit quantum channels, analyzes their classical restrictions, and studies the distribution of state transformations under random quantum channels.

Contribution

It provides explicit formulas for the volume of qubit channels, analyzes their structure via Choi representation, and examines state transformations under random channels.

Findings

Computed volume of general and unital qubit channels.

Derived distribution formulas for quantum states after random channel application.

Analyzed the structure of quantum channels over fixed classical channels.

Abstract

The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze 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 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. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.