Optimal Bounds on Functions of Quantum States under Quantum Channels

TL;DR

This paper derives optimal bounds for functions measuring the distance or similarity between quantum states after applying various classes of quantum channels, providing fundamental limits in quantum information processing.

Contribution

It introduces a comprehensive framework for determining the best possible bounds of quantum state functions under different quantum channel classes, extending previous results.

Findings

Established bounds for trace norm, fidelity, and relative entropy under unitary channels.

Extended bounds to mixed unitary, unital, and all quantum channels.

Provided a unified approach for analyzing quantum state transformations.

Abstract

Let $\rho_1, \rho_2$ be quantum states and $(\rho_1,\rho_2) \mapsto D(\rho_1, \rho_2)$ be a scalar function such as the trace norm, the fidelity, and the relative entropy, etc. We determine optimal bounds for $D(\rho_1, \Phi(\rho_2))$ for $\Phi \in \mathcal{S}$ for different class of functions $D(\cdot, \cdot)$, where $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.