# Entropy of a Quantum Channel: Definition, Properties, and Application

## Document Type

Conference Proceeding

## Publication Date

6-1-2020

## Abstract

The von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the "relative entropy distance" between ρ and the maximally mixed state, here we define the entropy of a channel {N} as the difference of the number of physical qubits of the channel output with the "relative entropy distance" between {N} and the completely depolarizing channel. We establish that this definition satisfies all of the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.

## Publication Source (Journal or Book title)

IEEE International Symposium on Information Theory - Proceedings

## First Page

1903

## Last Page

1908

## Recommended Citation

Gour, G., & Wilde, M.
(2020). Entropy of a Quantum Channel: Definition, Properties, and Application.* IEEE International Symposium on Information Theory - Proceedings**, 2020-June*, 1903-1908.
https://doi.org/10.1109/ISIT44484.2020.9174135