  We study restrictions imposed by quantum mechanics on the process of matrix
elements transfer. This problem is at the core of quantum measurements and
state transfer. Given two systems $\A$ and $\B$ with initial density matrices
$\lambda$ and $r$, respectively, we consider interactions that lead to
transferring certain matrix elements of unknown $\lambda$ into those of the
final state ${\widetilde r}$ of $\B$. We find that this process eliminates the
memory on the transferred (or certain other) matrix elements from the final
state of $\A$. If one diagonal matrix element is transferred, ${\widetilde
r}_{aa}=\lambda_{aa}$, the memory on each non-diagonal element
$\lambda_{a\not=b}$ is completely eliminated from the final density operator of
$\A$. Consider the following three quantities $\Re \la_{a\not =b}$, $\Im
\la_{a\not =b}$ and $\la_{aa}-\la_{bb}$ (the real and imaginary part of a
non-diagonal element and the corresponding difference between diagonal
elements). Transferring one of them, e.g., $\Re\tir_{a\not = b}=\Re\la_{a\not =
b}$, erases the memory on two others from the final state of $\A$.
Generalization of these set-ups to a finite-accuracy transfer brings in a
trade-off between the accuracy and the amount of preserved memory. This
trade-off is expressed via system-independent uncertainty relations which
account for local aspects of the accuracy-disturbance trade-off in quantum
measurements.
