Transferring elements of a density matrix

TL;DR

This paper investigates the fundamental quantum restrictions on transferring matrix elements between systems, revealing how such processes erase memory of certain elements and establishing uncertainty relations for the accuracy-memory trade-off.

Contribution

It introduces a detailed analysis of matrix element transfer restrictions in quantum mechanics and derives uncertainty relations for the accuracy-memory trade-off in quantum measurements.

Findings

Transferring a diagonal element erases memory of off-diagonal elements.

Memory of certain matrix elements is eliminated after transfer.

Uncertainty relations quantify the trade-off between transfer accuracy and memory preservation.

Abstract

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.