Transferring elements of a density matrix
ARMEN E. ALLAHVERDYAN, Karen V. Hovhannisyan · Physical Review A · 2010
We study restrictions imposed by quantum mechanics on the process of matrix-element transfer. This problem is at the core of quantum measurements and state transfer. Given two systems $A$ and $B$ with initial density matrices $\ensuremath{\lambda}$ and $r$, respectively, we consider interactions that lead to transferring certain matrix elements of unknown $\ensuremath{\lambda}$ into those of the final state $\mathrm{r\ifmmode \tilde{}\else \~{}\fi{}}$ 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, ${\mathrm{r\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{aa}}={\ensuremath{\lambda}}_{\mathit{aa}}$, the memory on each nondiagonal element ${\ensuremath{\lambda}}_{a\ensuremath{ e}b}$ is completely eliminated from the final density operator of $A$. Consider the following three quantities, $\mathrm{Re}{\ensuremath{\lambda}}_{a\ensuremath{ e}b}$, $\mathrm{Im}{\ensuremath{\lambda}}_{a\ensuremath{ e}b}$, and ${\ensuremath{\lambda}}_{\mathit{aa}}\ensuremath{-}{\ensuremath{\lambda}}_{\mathit{bb}}$ (the real and imaginary part of a nondiagonal element and the corresponding difference between diagonal elements). Transferring one of them, e.g., $\mathrm{Re}{\mathrm{r\ifmmode \tilde{}\else \~{}\fi{}}}_{a\ensuremath{ e}b}=\mathrm{Re}{\ensuremath{\lambda}}_{a\ensuremath{ e}b}$, erases the memory on two others from the final state of $A$. Generalization of these setups 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 that account for local aspects of the accuracy-disturbance trade-off in quantum measurements. Thus, the general aspect of state disturbance in quantum measurements is elimination of memory on non-diagonal elements, rather than diagonalization.