(in English) |
Consider a classical memory system, which is in the state described by
probability distribution $p$ or $q$ depending on the value of the bit
recorded. Suppose we embed this information into quantum system, $rho
=Gamma(p)$ or $sigma=Gamma(q)$.
Intuitively, the distribution ${p',q'}$ of measurement outcome on
${rho,sigma}$ should contain strictly less information than ${p,q}$
provided ${rho,sigma}$ is non-commutative. The statement holds if
the information is measured by relative entropy, Renyi relative entropy etc.
However, the assertion is untrue for the total variation distance $Vert
p-qVert_1$: if ${rho,sigma}$ satisfies some not very restrictive
conditions, $Vert p^{prime}-q^{prime}Vert_1$ equals $Vert p-qVert_1%
$. Here we present sufficient condition for general dimension, and necessary
and sufficient condition for qubit. |