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arXiv:quant-ph/0011057 · Journal of Physics A Mathematical and General

Optimal phase estimation and square root measurement

Masahide Sasaki, A. Carlini, Anthony Chefles

Published Aug 24, 2001• 5 citations
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Abstract

We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite-dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so-called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two-dimensional system.

DOI
10.1088/0305-4470/34/35/327

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We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite-dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so-called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two-dimensional system.
We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite dimensional system using a finite number of identical samples. Assuming the uniform {\\it a priori} distribution for the displacement parameter, an optimal strategy can be constructed by making the {\\it square root measurement} based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two dimensional system.
The theory of quantum state estimation is exploited here to investigate the most efficient strategies for this task, especially targeting a complete picture identifying optimal conditions in terms of Fisher information, quantum measurement, and associated estimator. The approach is specified to estimation of the phase of a qubit in a rotation around an arbitrary given axis, equivalent to estimating the phase of an arbitrary single-qubit quantum gate, both in noise-free and then in noisy conditions. In noise-free conditions, we establish the possibility of defining an optimal quantum probe, optimal quantum measurement, and optimal estimator together capable of achieving the ultimate best performance uniformly for any unknown phase. With arbitrary quantum noise, we show that in general the optimal solutions are phase dependent and require adaptive techniques for practical implementation. However, for the important case of the depolarizing noise, we again establish the possibility of a quantum probe, quantum measurement, and estimator uniformly optimal for any unknown phase. In this way, for qubit phase estimation, without and then with quantum noise, we characterize the phase-independent optimal solutions when they generally exist, and also identify the complementary conditions where the optimal solutions are phase dependent and only adaptively implementable.
We provide the optimal measurement strategy for a class of noisy channels that reduce to the identity channel for a specific value of a parameter (spreading channels). We provide an example that is physically relevant: the estimation of the absolute value of the displacement in the presence of phase randomizing noise. Surprisingly, this noise does not affect the effectiveness of the optimal measurement. We show that, for small displacement, a squeezed vacuum probe field is optimal among strategies with same average energy. A squeezer followed by photodetection is the optimal detection strategy that attains the quantum Fisher information, whereas the customarily used homodyne detection becomes useless in the limit of small displacements, due to the same effect that gives Rayleigh's curse in optical superresolution. There is a quantum advantage: a squeezed or a Fock state with N average photons allow to asymptotically estimate the parameter with a sqrt[N] better precision than classical states with same energy.
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of NNN independent and identically prepared two-level systems (qubits).
The optimal-phase estimation strategy is derived when partial a priori knowledge on the estimated phase is available. The solution is found with the help of the most famous result from the entanglement theory: the positive partial transpose criterion. The structure of the optimal measurements, estimators, and the optimal probe states is analyzed. This Rapid Communication provides a unified framework bridging the gap in the literature on the subject which until now dealt almost exclusively with two extreme cases: almost perfect knowledge (local approach based on Fisher information) and no a priori knowledge (global approach based on covariant measurements). Special attention is paid to a natural a priori probability distribution arising from a diffusion process.