Average output entropy for quantum channels

TL;DR

This paper investigates the regularized average Renyi output entropy of quantum channels, providing explicit formulas for certain channels and conjecturing their equality with related quantities, revealing non-analytic behavior in some cases.

Contribution

It derives a closed-form expression for the regularized average Renyi output entropy and conjectures its equality with a related quantity, including explicit results for specific quantum channels.

Findings

Explicit formula for $eta_{r}^{ eg}$ for some channels

Equality of $eta_{r}^{ eg}$ and $ar{S}_{r}^{ eg}$ in certain cases

Non-analytic behavior of the entropy functions for the depolarizing channel

Abstract

We study the regularized average Renyi output entropy $\bar{S}_{r}^{\reg}$ of quantum channels. This quantity gives information about the average noisiness of the channel output arising from a typical, highly entangled input state in the limit of infinite dimensions. We find a closed expression for $\beta_{r}^{\reg}$, a quantity which we conjecture to be equal to $\Srreg$. We find an explicit form for $\beta_{r}^{\reg}$ for some entanglement-breaking channels, and also for the qubit depolarizing channel $\Delta_{\lambda}$ as a function of the parameter $\lambda$. We prove equality of the two quantities in some cases, in particular we conclude that for $\Delta_{\lambda}$ both are non-analytic functions of the variable $\lambda$.