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arXiv:0902.3260

Classical Fisher Information in Quantum Metrology -- Interplay of Probe, Dynamics and Measurement

Gabriel A. Durkin

Published Feb 18, 2009• 0 citations
arXivHTMLPDF

Abstract

We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which establishes the optimal precision for a fixed input. The formalism demonstrates that, in general, the classical Fisher Information has the Hamiltonian semi-norm as an upper bound. This is achievable with a qubit probe and only projective measurements, and is independent of the true value of the estimated parameter. In an interferometry context, we show that an optimal measurement scheme can be constructed from linear optics and photon counting, without recourse to generalised measurements or exotic unitaries outside of SU(2).

DOI
10.48550/arxiv.0902.3260

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We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which establishes the optimal precision for a fixed input. The formalism demonstrates that, in general, the classical Fisher Information has the Hamiltonian semi-norm as an upper bound. This is achievable with a qubit probe and only projective measurements, and is independent of the true value of the estimated parameter. In an interferometry context, we show that an optimal measurement scheme can be constructed from linear optics and photon counting, without recourse to generalised measurements or exotic unitaries outside of SU(2).
We explore precision in a measurement process incorporating pure probe states, unitary dynamics and complete measurements via a simple formalism. The concept of 'information complement' is introduced. It undermines measurement precision and its minimization reveals the optimal system properties. Maximally precise measurements can exhibit independence from the true value of the estimated parameter, but demanding this severely restricts the type of viable probe and dynamics, including the requirement that the Hamiltonian be block diagonal in a basis of preferred measurements. The curvature of the information complement near a globally optimal point provides a new quantification of measurement stability.
We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These results make precise and reveal the full potential of mixed states for quantum metrology.
Measurement noise is a major source of noise in quantum metrology.Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision.We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols.First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases.Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases.Finally, we demonstrate that in multiprobe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements.Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls.
Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.
Quantum metrology is superior to classical metrology in parameter estimation. This is because quantum states have coherence that, however, does not exist in classical states. Previous works were limited to the study of the relationship between quantum Fisher information (QFI) and coherence of the initial probe state only for a specific single-qubit parameter estimation process (e.g., a process described by a single-qubit spin angular momentum operator Jz{J}_{z}Jz​). Here, we consider a more general case and study the relationship between QFI and coherence of the initial probe state for a single-qubit general parameter estimation process, which is described by a single-qubit spin angular momentum operator ${J}_{\stackrel{P\vec}{n}}$ with a general unit vector $\stackrel{P\vec}{n}$. We find that the QFI is proportional to the square of coherence of the initial probe state, when the single-qubit bases are eigenstates of the spin angular momentum operator ${J}_{\stackrel{P\vec}{n}}$ describing the parametrization process. We also design and conduct a linear optical experiment to support our theory. The experimental results are in good agreement with the theoretical ones. This work provides an important guideline for enhancing the precision of parameter estimation.
Fisher information is a statistical metric whose inverse represents a lower bound on the variance of an unbiased estimator. Naturally, in order to make the best estimates of a given unknown parameter, one would desire to find ways to minimize the value of this lower bound. Thus, the Fisher information can also be used to compare the quality of different estimators for a given parameter: whichever estimator has a lower value for its Fisher information must also provide better estimates. The same holds true in the framework of quantum information theory, where an analog to the Fisher information, dubbed quantum Fisher information, performs the same role as its classical counterpart. This dissertation concerns itself with both types of Fisher information, and so I present two projects in which Fisher information was a helpful tool for determining the quality of our work. In the first project, we examined potential solutions to the problem of optical loss in long-baseline interferometers, which creates a bottleneck in the resolution these arrays of telescopes can achieve. What we suggested was a novel entanglement-assisted scheme that interferes a two-mode squeezed vacuum state with the target astronomical state, and then measures the resulting state by means of homodyne detection. Ultimately, we showed that when the Fisher information is observed, our proposed scheme did not outperform the traditional schemes used in long-baseline interferometry. In the second project, we sought to write code that could determine how to best use dynamical decoupling pulses to correct dephasing error in a nitrogen-vacancy center realization of a memory and spectator qubit system. Through the use of the Python coding language, we were able to write programs that could determine the characteristic time and model the change in fidelity of the system when XY4 pulses are applied. Ultimately, we found that the quantum fidelity of our model closely mirrored the fidelity decay function our system was expected to follow in the case where XY4 pulses are applied.