  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.
