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

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.

References in corpus (1)

• A new quantum version of f-divergence