# An information-theoretic treatment of quantum dichotomies

## TL;DR

This paper develops an information-theoretic framework for quantum state transformations, showing that quantum relative entropy determines the feasibility and rate of such transformations under small errors and asymptotic limits.

## Contribution

It introduces a simplified condition based on one-shot relative entropies for approximate transformations and establishes the quantum relative entropy as the key quantity in asymptotic state transformation rates.

## Key findings

- Quantum relative entropy governs transformation feasibility.
- Exponential error decay when the initial entropy exceeds the target.
- Transformation rate equals the ratio of relative entropies.

## Abstract

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence.