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Journal of Physics A Mathematical and General

Quantum parameter estimation of a generalized Pauli channel

Akio Fujiwara, Hiroshi Imai

Published Jul 9, 2003• 109 citations
Publisher

Abstract

We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : ( d ) → ( d ), where the parameter θ is regarded as a coordinate system of the probability simplex d 2 −1 . We show that for each degree n of extension (id ⊗ Γ θ ) ⊗ n : (( d ⊗ d ) ⊗ n ) → (( d ⊗ d ) ⊗ n ), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n -tensor product of a maximally entangled state τ ME ∊ ( d ⊗ d ). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇ m -affine in d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ ME )} θ of output states can be identified with d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.

DOI
10.1088/0305-4470/36/29/314

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We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : ( d ) → ( d ), where the parameter θ is regarded as a coordinate system of the probability simplex d 2 −1 . We show that for each degree n of extension (id ⊗ Γ θ ) ⊗ n : (( d ⊗ d ) ⊗ n ) → (( d ⊗ d ) ⊗ n ), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n -tensor product of a maximally entangled state τ ME ∊ ( d ⊗ d ). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇ m -affine in d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ ME )} θ of output states can be identified with d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.
A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.
The optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/ N , where N is the number of used measurements.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when
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 optimal measurement configuration, i.e. the optimal input quantum state and measurement in the form of a positive operator-valued measure with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the appropriate channel directions. Extensions of these results for generalized Pauli channels are also given. Furthermore, a computationally efficient estimation method is proposed for estimating the channel directions in the qubit Pauli channel case. The efficiency of this method is illustrated by simulations: it is shown that its variance is of order 1/N, where N is the number of used measurements.
A fibre bundle structure is introduced over manifolds of quantum channels. This structure has a close connection with the problem of estimating an unknown quantum channel Γ θ specified by a parameter θ. It is shown that the quantum Fisher information of the family of output states maximized over all input states , which quantifies the ultimate statistical distinguishability of the parameter θ, is expressed in terms of a geometrical quantity on the fibre bundle. Using this formula, a criterion for the maximum quantum Fisher information of the n th extended channel (id ⊗ Γ θ ) ⊗ n to be O ( n ) is derived. This criterion further proves that for almost all quantum channels, the maximum quantum Fisher information increases in the order of O ( n ).