Reverse-type Data Processing Inequality

TL;DR

This paper investigates reverse quantum data processing inequalities by analyzing contraction and expansion coefficients of quantum channels, revealing conditions under which distinguishability is preserved or expanded, and constructing specific non-degradable channels.

Contribution

It introduces the concept of a relative expansion coefficient for quantum channels and demonstrates its positivity for key channel classes, also constructing non-degradable channels.

Findings

Channels with input dimension ≥ output dimension lack a non-zero expansion coefficient.

Relative expansion coefficient is positive for depolarizing, dephasing, and amplitude damping channels.

Constructs the first examples of non-degradable, less noisy quantum channels.

Abstract

The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.