We combine the long-distance quantum state transfer and simple operations with the elements of the transferred (nor perfectly) density matrix. These operations are turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. The basic tool performing these operations is the unitary transformation on the extended receiver. A system of linear algebraic equations can be solved in this way as well. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.
The evolution of quantum coherences comes with a set of conservation laws provided that the Hamiltonian governing this evolution conserves the spin-excitation number. At that, coherences do not intertwist during the evolution. Using the transmission line and the receiver in the initial ground state we can transfer the coherences to the receiver without interaction between them, although the matrix elements contributing to each particular coherence intertwist in the receiver’s state. Therefore we propose a tool based on the unitary transformation at the receiver side to untwist these elements and thus restore (at least partially) the structure of the sender’s initial density matrix. A communication line with two-qubit sender and receiver is considered as an example of implementation of this technique.
Abstract Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the ‘sender-receiver’ model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver’s density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.
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 ${\ensuremath{\lambda}}_{a\ensuremath{\ne}b}$ is completely eliminated from the final density operator of
A. Consider the following three quantities, $\mathrm{Re}{\ensuremath{\lambda}}_{a\ensuremath{\ne}b}$, $\mathrm{Im}{\ensuremath{\lambda}}_{a\ensuremath{\ne}b}$, 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., $\mathrm{Re}{\mathrm{r\ifmmode \tilde{}\else \~{}\fi{}}}_{a\ensuremath{\ne}b}=\mathrm{Re}{\ensuremath{\lambda}}_{a\ensuremath{\ne}b}$, 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.