Quantum restrictions on transfer of matrix elements

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.