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arXiv:0811.2528

Quantum restrictions on transfer of matrix elements

ARMEN E. ALLAHVERDYAN, Karen V. Hovhannisyan

Published Nov 15, 2008• 0 citations
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Abstract

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 $λ$ and $r$, respectively, we consider most general interactions, which lead to transferring certain matrix elements of unknown $λ$ 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: ${\widetilde r}_{aa}=λ_{aa}$, the memory on each non-diagonal element $λ_{a\not=b}$ is completely eliminated from the final density operator of A. The transfer of a non-diagonal element: ${\widetilde r}_{ab}=λ_{ab}$ eliminates the memory on the diagonal elements $λ_{aa}$ and $λ_{bb}$, while the memory about their sum $λ_{aa}+λ_{bb}$ is kept. Moreover, the memory about $λ_{ab}$ 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.

DOI
10.48550/arxiv.0811.2528

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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 λλλ and rrr, respectively, we consider most general interactions, which lead to transferring certain matrix elements of unknown λλλ into those of the final state r~{\widetilde r}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: r~aa=λaa{\widetilde r}_{aa}=λ_{aa}raa​=λaa​, the memory on each non-diagonal element λa≠bλ_{a\not=b}λa=b​ is completely eliminated from the final density operator of A. The transfer of a non-diagonal element: r~ab=λab{\widetilde r}_{ab}=λ_{ab}rab​=λab​ eliminates the memory on the diagonal elements λaaλ_{aa}λaa​ and λbbλ_{bb}λbb​, while the memory about their sum λaa+λbbλ_{aa}+λ_{bb}λaa​+λbb​ is kept. Moreover, the memory about λabλ_{ab}λab​ 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.
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 AAA and BBB with initial density matrices $\ensuremath{\lambda}$ and rrr, 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 BBB. We find that this process eliminates the memory on the transferred (or certain other) matrix elements from the final state of AAA. 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 AAA. 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 AAA. 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.
Using the tool of unitary transformations of the extended receiver we perform simple operations with the non-diagonal elements of the initial sender's density matrix after their transferring to the receiver. These operations are following: turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. We also propose a protocol for solving a system of linear algebraic equations with a system of two equations as an example. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.
Any physical process can be represented as a quantum channel mapping an initial state to a final state. Hence it can be characterized from the point of view of communication theory, i.e., in terms of its ability to transfer information. Quantum information provides a theoretical framework and the proper mathematical tools to accomplish this. In this context the notion of codes and communication capacities have been introduced by generalizing them from the classical Shannon theory of information transmission and error correction. The underlying assumption of this approach is to consider the channel not as acting on a single system, but on sequences of systems, which, when properly initialized allow one to overcome the noisy effects induced by the physical process under consideration. While most of the work produced so far has been focused on the case in which a given channel transformation acts identically and independently on the various elements of the sequence (memoryless configuration in jargon), correlated error models appear to be a more realistic way to approach the problem. A slightly different, yet conceptually related, notion of correlated errors applies to a single quantum system which evolves continuously in time under the influence of an external disturbance which acts on it in a non-Markovian fashion. This leads to the study of memory effects in quantum channels: a fertile ground where interesting novel phenomena emerge at the intersection of quantum information theory and other branches of physics. A survey is taken of the field of quantum channels theory while also embracing these specific and complex settings.
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.
Informationally complete measurements allow the estimation of expectation values of any operator on a quantum system, by changing only the data processing of the measurement outcomes. In particular, an informationally complete measurement can be used to perform quantum tomography, namely to estimate the density matrix of the quantum state. The data processing is generally nonunique, and can be optimized according to a given criterion. In this paper we provide the solution of the optimization problem which minimizes the variance in the estimation. We then consider informationally complete measurements performed over bipartite quantum systems focusing attention on universally covariant measurements, and compare their statistical efficiency when performed either locally or globally on the two systems. Among global measurements we consider the special case of Bell measurements, which allow us to estimate the expectation of a restricted class of operators. We compare the variance in the three cases: local, Bell, and unrestricted global---and derive conditions for the operators to be estimated such that one type of measurement is more efficient than the other. In particular, we find that for factorized operators and Bell projectors the Bell measurement always performs better than the unrestricted global measurement, which in turn outperforms the local one. For estimation of the matrix elements of the density operator, the relative performances depend on the basis on which the state is represented, and on the matrix element being diagonal or off-diagonal, however, with the global unrestricted measurement generally performing better than the local one.