We consider a two-mode bosonic state with fixed photon number
n∈N , whose upper and lower modes pick up a phase
ϕ and
−ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating
ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider
ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider
n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE over all possible physically allowable measurements and the resulting MMSE (Personick, 1997). This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in aBayesiansetting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multi-parameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.
In Bayesian estimation theory, the estimator
θ^=E[θ∣l] attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest
θ from the observation of
l through a noisy channel
Pl∣θ, given a prior
Pθ on
θ. In quantum sensing tasks, the user gets
ρθ, the quantum state that encodes
θ. They choose a measurement, a positive-operator valued measure (POVM)
Πl, which induces the channel
Pl∣θ=Tr(ρθΠl) to the measurement outcome
l, on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM
Πl that minimizes the MMSE over all possible measurements, and that MMSE. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements, when
Pθ is unavailable. For multi-parameter estimation, i.e., when
θ is a vector, in Fisher quantum estimation theory, the inverse of the QFI matrix provides an operator lower bound to the covariance of an unbiased estimator. However, there has been little work on quantifying quantum limits and measurement designs, for multi-parameter quantum estimation in the {\em Bayesian} setting. In this paper, we build upon Personick's result to construct a Bayesian adaptive measurement scheme for multi-parameter estimation when
N copies of
ρθ are available. We illustrate an application to localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs far superior to quantum-noise-limited focal-plane direct imaging.
We consider the problem of estimating an unknown but constant carrier phase modulation $\ensuremath{\theta}$ using a general, possibly entangled,
n-mode optical probe through
n independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\ensuremath{\theta}$ as a function of
n, the mean and variance of the total number of photons
NS in the
n-mode probe, the transmissivity $\ensuremath{\eta}$, and mean thermal photon number per mode
nB of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-square error (MSE) of an unbiased estimator $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\theta}}$ of $\ensuremath{\theta}$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.
We examine the problem of estimating the separation between two arbitrarily close incoherent point sources from a quantum Bayesian perspective—that is, assuming the availability of a prior probability distribution function (PDF) for the separation. In particular, we consider both non-displaced and displaced half-Gaussian prior PDFs to evaluate the performance of Hermite–Gauss spatial mode demultiplexing (HG-SPADE) and direct imaging (DI). By employing the Bayesian minimum mean square error criterion, which is always attainable by a different kind of spatial demultiplexing, we compare the effectiveness of HG-SPADE and DI and analyze how variations in the parameters of the prior PDF influence the conditions under which one method outperforms the other.
Single photon count distributions at the outputs of a Mach-Zehnder interferometer are highly dependent on the phase difference between the two arms of the apparatus, so photocount records can be used to estimate an unknown optical phase. Experimental results have shown that phase information can be inferred from the data even at low mean input intensities of ⪅10 photons. Here we consider the optimal Bayesian parameter estimate for the model used by a previous frequentist analysis. In the limit of high input intensity the observation is well-modelled as an intensity measurement of a classical wave in Gaussian measurement noise, however in the low intensity regime this simple model does not apply. It seems that no generally valid closed-form estimator for this problem exists without making the Gaussian approximation. Moreover the 2π radian ambiguity in phase estimates presents a problem in defining cost functions to compare estimator performance, and we consider two reasonable alternatives. We present a numerical study of the experiment along with a novel numerical approach to this problem which out-performs the existing estimators and with a squared error that approaches the Cramer-Rao lower bound previously derived in the literature, in the limit where that bound is valid.