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arXiv:1712.07858 · Physical Review A

Estimation of general Hamiltonian parameters via controlled energy measurements

Luigi Seveso, Matteo G. A. Paris

Published Sep 20, 2018• 11 citations
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Abstract

The quantum Cram\'er-Rao theorem states that the quantum Fisher information bounds the best achievable precision in the estimation of a quantum parameter $\ensuremath{\xi}$. This is true, however, under the assumption that the measurement employed to extract information on $\ensuremath{\xi}$ is regular, i.e., neither its sample space nor its positive-operator valued elements depend on the (true) value of the parameter. A better performance may be achieved by relaxing this assumption. In the case of a general Hamiltonian parameter, i.e., when the parameter enters the system's Hamiltonian in a nonlinear way (making the energy eigenstates and eigenvalues $\ensuremath{\xi}$ dependent), a family of nonregular measurements, referred to as controlled energy measurements, is naturally available. We perform an analytic optimization of their performance, which enables us to compare the optimal controlled energy measurement with the optimal Braunstein-Caves measurement based on the symmetric logarithmic derivative. As the former may outperform the latter, the ultimate quantum bounds for general Hamiltonian parameters are different than those for phase (shift) parameters. We also discuss in detail a realistic implementation of controlled energy measurements based on the quantum phase estimation algorithm and work out a variety of examples to illustrate our results.

DOI
10.1103/physreva.98.032114

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The quantum Cram'er-Rao theorem states that the quantum Fisher information bounds the best achievable precision in the estimation of a quantum parameter \ensuremathξ\ensuremath{\xi}\ensuremathξ. This is true, however, under the assumption that the measurement employed to extract information on \ensuremathξ\ensuremath{\xi}\ensuremathξ is regular, i.e., neither its sample space nor its positive-operator valued elements depend on the (true) value of the parameter. A better performance may be achieved by relaxing this assumption. In the case of a general Hamiltonian parameter, i.e., when the parameter enters the system's Hamiltonian in a nonlinear way (making the energy eigenstates and eigenvalues \ensuremathξ\ensuremath{\xi}\ensuremathξ dependent), a family of nonregular measurements, referred to as controlled energy measurements, is naturally available. We perform an analytic optimization of their performance, which enables us to compare the optimal controlled energy measurement with the optimal Braunstein-Caves measurement based on the symmetric logarithmic derivative. As the former may outperform the latter, the ultimate quantum bounds for general Hamiltonian parameters are different than those for phase (shift) parameters. We also discuss in detail a realistic implementation of controlled energy measurements based on the quantum phase estimation algorithm and work out a variety of examples to illustrate our results.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.
The attainability of the quantum Cram\'er-Rao bound (QCR), the ultimate limit in the precision of the estimation of a physical parameter, requires the saturation of the quantum information bound (QIB). This occurs when the Fisher information associated to a given measurement on the quantum state of a system which encodes the information about the parameter coincides with the quantum Fisher information associated to that quantum state. Braunstein and Caves [Phys. Rev. Lett. 72, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called the symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated, therefore requiring previous knowledge of the quantity one is trying to estimate. For this reason, it is important to investigate under which situation it is possible to saturate the QCR without previous information about the parameter to be estimated. Here, we show the complete solution to the problem of which are the initial pure states and which projective measurements allow the global saturation of the QIB, without the knowledge of the true value of the parameter, when the information about the parameter is encoded in the system by a unitary process.
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.