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Physical Review A

Quantum channel identification problem

Akio Fujiwara

Published Mar 16, 2001• 195 citations
Publisher

Abstract

This paper explores an application of quantum entanglement. The problem treated here is the quantum channel identification problem: given a parametric family ${{\ensuremath{\Gamma}}_{\ensuremath{\theta}}{}}_{\ensuremath{\theta}}$ of quantum channels, find the best strategy of estimating the true value of the parameter \ensuremath{\theta}. As a simple example, we study the estimation problem of the isotropic depolarization parameter \ensuremath{\theta} for a two-level quantum system $\mathcal{H}\ensuremath{\simeq}{\mathbf{C}}^{2}.$ In the framework of noncommutative statistics, it is shown that the optimal input state on $\mathcal{H}\ensuremath{\bigotimes}\mathcal{H}$ to the channel exhibits a transitionlike behavior according to the value of the parameter \ensuremath{\theta}.

DOI
10.1103/physreva.63.042304

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This paper explores an application of quantum entanglement. The problem treated here is the quantum channel identification problem: given a parametric family \ensuremathΓ\ensuremathθ\ensuremathθ{{\ensuremath{\Gamma}}_{\ensuremath{\theta}}{}}_{\ensuremath{\theta}}\ensuremathΓ\ensuremathθ​\ensuremathθ​ of quantum channels, find the best strategy of estimating the true value of the parameter \ensuremath{\theta}. As a simple example, we study the estimation problem of the isotropic depolarization parameter \ensuremath{\theta} for a two-level quantum system H\ensuremath≃C2.\mathcal{H}\ensuremath{\simeq}{\mathbf{C}}^{2}.H\ensuremath≃C2. In the framework of noncommutative statistics, it is shown that the optimal input state on H\ensuremath⨂H\mathcal{H}\ensuremath{\bigotimes}\mathcal{H}H\ensuremath⨂H to the channel exhibits a transitionlike behavior according to the value of the parameter \ensuremath{\theta}.
Statistical estimation and test of unknown channels have attracted interest of many researchers. In optimizing the process of inference, an important step is optimization of the input state, which in general do depend on the kind of inference (estimation or test, etc.), on the error measure, and so on. But sometimes, there is a universally optimal input state, or an input state best for all the statistical inferences and for all the risk functions. In the paper, the existence of a universally optimal state is shown for group covariant/contravariant channels, unital qubit channels and some measurement families. To prove these results, theory of "comparison of state families" are used. We also discuss about effectiveness of entanglement and adaptation of input states.
Suppose that three kinds of quantum systems are given in some unknown states $|f{〉}^{\ensuremath{\bigotimes}N},$ $|{g}_{1}{〉}^{\ensuremath{\bigotimes}K},$ and $|{g}_{2}{〉}^{\ensuremath{\bigotimes}K},$ and we want to decide which template state ∣g1〉|{g}_{1}〉∣g1​〉 or ∣g2〉,|{g}_{2}〉,∣g2​〉, each representing the feature of the pattern class C1{\mathcal{C}}_{1}C1​ or C2,{\mathcal{C}}_{2},C2​, respectively, is closest to the input feature state ∣f〉.|f〉.∣f〉. This is an extension of the pattern matching problem into the quantum domain. Assuming that these states are known a priori to belong to a certain parametric family of pure qubit systems, we derive two kinds of matching strategies. The first one is a semiclassical strategy that is obtained by the natural extension of conventional matching strategies and consists of a two-stage procedure: identification (estimation) of the unknown template states to design the classifier (learning process to train the classifier) and classification of the input system into the appropriate pattern class based on the estimated results. The other is a fully quantum strategy without any intermediate measurement, which we might call as the universal quantum matching machine. We present the Bayes optimal solutions for both strategies in the case of K=1,K=1,K=1, showing that there certainly exists a fully quantum matching procedure that is strictly superior to the straightforward semiclassical extension of the conventional matching strategy based on the learning process.
For any $N\ensuremath{\bigotimes}N$-type bipartite system with even N>2N>2N>2, two-parameter continuous families of unital block-dephasing quantum channels and of weakly optimal entanglement witnesses are constructed from special forms of a parametrized positive map. The parameter domains of the families are complementary subsets of a convex set and this fact reveals unusual combination rules of quantum channels and entanglement witnesses. These rules make it possible to develop many physically implementable direct entanglement-detection methods that provide a unified framework to accomplish both information-processing and entanglement-detection tasks by means of smooth quantum channels, without any reference to entanglement witnesses.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
The system identification problem is to estimate dynamical parameters from the output data, obtained by performing measurements on the output fields. We investigate system identification for quantum linear systems. Our main objectives are to address the following general problems: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realisation from sufficient input-output data? (3) How well can we estimate the parameters governing the dynamics? We investigate these problems in two contrasting approaches; using time-dependent inputs or time-stationary (quantum noise) inputs.