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arXiv:0807.3958 · Physical Review A

Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states

Gerardo Adesso, Fabio Dell’Anno, Silvio De Siena, Fabrizio Illuminati, Leonardo A. M. Souza

Published Apr 23, 2009• 180 citations
arXivHTMLPDF

Abstract

We address the estimation of the loss parameter of a bosonic channel probed by arbitrary signals. Unlike the optimal Gaussian probes, which can attain the ultimate bound on precision asymptotically either for very small or very large losses, we prove that Fock states at any fixed photon number saturate the bound unconditionally for any value of the loss. In the relevant regime of low-energy probes, we demonstrate that superpositions of the first low-lying Fock states yield an absolute improvement over any Gaussian probe. Such few-photon states can be recast quite generally as truncations of de-Gaussified photon-subtracted states.

DOI
10.1103/physreva.79.040305

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We address the estimation of the loss parameter of a bosonic channel probed by arbitrary signals. Unlike the optimal Gaussian probes, which can attain the ultimate bound on precision asymptotically either for very small or very large losses, we prove that Fock states at any fixed photon number saturate the bound unconditionally for any value of the loss. In the relevant regime of low-energy probes, we demonstrate that superpositions of the first low-lying Fock states yield an absolute improvement over any Gaussian probe. Such few-photon states can be recast quite generally as truncations of de-Gaussified photon-subtracted states.
We address the estimation of the loss parameter of a bosonic channel probed by Gaussian signals. We derive the ultimate quantum bound with precision and show that no improvement may be obtained by having access to the environmental degrees of freedom. We find that, for small losses, the variance of the optimal estimator is proportional to the loss parameter itself, a result that represents a qualitative improvement over the shot-noise limit. An observable based on the symmetric logarithmic derivative is obtained, which attains the ultimate bound and may be implemented using Gaussian operations and photon counting.
We address the characterization of dissipative bosonic channels and show that estimation of the loss rate by Gaussian probes (coherent or squeezed) is improved in the presence of Kerr nonlinearity. In particular, enhancement of precision may be substantial for short interaction time, i.e., for media of moderate size, e.g., biological samples. We analyze in detail the behavior of the quantum Fisher information (QFI), and determine the values of nonlinearity maximizing the QFI as a function of the interaction time and of the parameters of the input signal. We also discuss the precision achievable by photon counting and quadrature measurement and present additional results for truncated, few-photon, probe signals. Finally, we discuss the origin of the precision enhancement, showing that it cannot be linked quantitatively to the non-Gaussianity or the nonclassicality of the interacting probe signal.
Abstract Quantum mechanics establishes the ultimate limit to the scaling of the precision on any parameter, by identifying optimal probe states and measurements. While this paradigm is, at least in principle, adequate for the metrology of quantum channels involving the estimation of phase and loss parameters, we show that estimating the loss parameters associated with a quantum channel and a realistic quantum detector are fundamentally different. While Fock states are provably optimal for the former, we identify a crossover in the nature of the optimal probe state for estimating detector imperfections as a function of the loss parameter using Fisher information as a benchmark. We provide theoretical results for on-off and homodyne detectors, the most widely used detectors in quantum photonics technologies, when using Fock states and coherent states as probes.
We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator’s deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a useful asymptotic characterization of identically prepared states in terms of Gaussian states. We first prove our results for the mean square error of a special class of models, called D -invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, i.e. parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.
We address the problem of estimating the transmissivity of the pure-loss single-mode bosonic channel from the Bayesian point of view, i.e., when a prior probability distribution function (PDF) on the transmissivity is available. We compute the quantum limit of the Bayesian minimum mean square error of estimating transmissivity. Specifically, we consider two prior PDFs: the two-point distribution (relevant for reading from an optical drive) and the beta distribution (relevant for imaging a reflective pixelated scene). If the probe's mean photon number is an integer, for the two-point PDF we prove that the optimal probe is a Fock state and the optimal measurement is photon counting, while for the beta PDF our numerical results provide evidence on the optimality of the Fock state probe and photon counting. When the probe's mean photon number is any (nonnegative) real number, we conjecture the form of the optimal probe state and we study the performance of photon counting, which is a suboptimal yet practical measurement. Our methods can be applied for any prior PDF. We discuss how different precision metrics and priors on the parameter influence the quantum-optimal probe and measurement. To our knowledge, these results on transmissivity estimation with quantum-limited precision represent the first Bayesian analysis of the problem.
Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed.We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons.The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved by a homodyne measurement on the single mode.We also reveal the best strategy for the estimation when we are given L identical target circuits and are allowed to apply passive linear controls in between with an arbitrary number of ancilla modes introduced.