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.
Categories: quant-ph math-ph math.MP