  We discuss restrictions imposed by quantum mechanics on the process of matrix
elements transfer from the one system to another. This is relevant for various
processes of partial state transfer (quantum communication, indirect
measurement, polarization transfer, {\it etc}). Given two systems A and B with
initial density operators $\lambda$ and $r$, respectively, we consider most
general interactions, which lead to transferring certain matrix elements of
unknown $\lambda$ into those of the final state ${\widetilde r}$ of B. We find
that this process leads to eliminating 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. The transfer of a non-diagonal element:
${\widetilde r}_{ab}=\lambda_{ab}$ eliminates the memory on the diagonal
elements $\lambda_{aa}$ and $\lambda_{bb}$, while the memory about their sum
$\lambda_{aa}+\lambda_{bb}$ is kept. Moreover, the memory about $\lambda_{ab}$
itself is completely eliminated from the final state of A. Generalization of
these set-ups to non-ideal transfer brings in a trade-off between the quality
of the transfer and the amount of preserved memory. This trade-off is expressed
via system-independent uncertainty relations.
