# Probability representation of quantum channels

## TL;DR

This paper develops a probability-based framework for representing quantum channels, linking quantum operations to classical probability distributions, and provides detailed examples for qubit states.

## Contribution

It introduces a novel probability representation of quantum channels, connecting completely positive maps with classical probability distributions for qudits.

## Key findings

- Quantum channels can be represented by sets of classical probability distributions.
- The evolution of quantum states can be described by classical-like kinetic equations.
- Explicit examples for qubit state maps are provided.

## Abstract

Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct the probability representation of the completely positive maps. In this representation, any completely positive map of qudit state density matrix is identified with the set of classical coin probability distributions. Examples of the maps of qubit states are studied in detail. The evolution equation of quantum states is written in the form of the classical-like kinetic equation for probability distributions identified with qudit state.