On the monotone metric of classical channel and distribution spaces: asymptotic theory

TL;DR

This paper characterizes a class of monotone metrics on the space of classical channels, revealing their non-Riemannian nature and proposing asymptotic assumptions to develop a geometric theory beyond traditional Riemannian metrics.

Contribution

It introduces asymptotic assumptions to analyze monotone channel metrics, extending geometric understanding beyond Riemannian frameworks.

Findings

Monotone channel metrics are proven not to be Riemannian.

Asymptotic assumptions enable new geometric characterizations.

Implications for quantum state metrics are discussed.

Abstract

The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.