# Reversibility of distance measures of states with some focus on total variation distance

## TL;DR

This paper investigates how classical probability distances relate to quantum states, showing that total variation distance can sometimes be preserved under measurement, contrary to previous assumptions, with specific conditions identified.

## Contribution

The paper extends the understanding of distance measure reversibility from operator convex functions to strictly convex functions and provides conditions under which total variation distance remains unchanged.

## Key findings

- Total variation distance can be preserved under measurement for certain quantum states.
- Extension of reversibility results to strictly convex functions beyond operator convex functions.
- Necessary and sufficient conditions identified for qubit states regarding total variation distance preservation.

## Abstract

Consider a classical system, which is in the state described by probability distribution $p$ or $q$, and embed these classical informations into quantum system by a physical map $\Gamma$, $\rho=\Gamma(p)$ and $\sigma=\Gamma(q)$. Intuitively, the pair $\{p_{\rho}^{M},p_{\sigma}^{M}\}$ of the distributions of the data of the measurement $M$ on the pair $\{\rho,\sigma\}$ should contain strictly less information than the pair $\{p,q\}$ provided the pair $\{\rho,\sigma\}$ is non-commutative. Indeed, this statement had been shown if the information is measured by $f$-divergence such that $f$ is operator convex. In the paper, the statement is extended to the case where $f$ is strictly convex. Also, we disprove the assertion for the total variation distance $\Vert p-q\Vert_{1}$, the $f$-divergence with $f(r)=|1-r|$: if $\{\rho,\sigma\}$ satisfies some not very restrictive conditions, $\Vert p_{\rho}^{M}-p_{\sigma}^{M}\Vert_{1}$ equals $\Vert p-q\Vert_{1}$. Here we present sufficient condition for general case, and necessary and sufficient condition for qubit states.