The problem of estimating the squared modulus of the mutual intensity (or the complex coherence factor) from high-light-level intensity measurements is addressed for the situation in which the fluctuations of the complex-valued amplitude are governed by circular-Gaussian statistics and the light level is high enough that all nonclassical fluctuations of the measurements can be ignored. A lower bound on the variance of any unbiased estimator is presented, and this bound is compared with the variance of Ebstein’s polynomial estimators [ J. Opt. Soc. Am. A8, 1450 ( 1991)] along with the variance of the maximum-likelihood estimator.
In many applications using light for practical purposes, the accuracy of optical experiment is limited most fundamentally by the finite amount of light utilized in the measurements. Special emphasis should be put on interferometry aimed at measuring the parameters of simple fringe patterns due to the following reasons. First, fringe parameter measurement provides a relatively well defined and tractable example of application of the theory. The desired parameters are easily defined, and methods for their measurement are readily devised based on common sense. Second, fringe parameter measurement is central to all problems involving coherence. The fundamental descriptors of light waves utilized in coherence theory are in fact measurable parameters of fringes. By examining the limitations to fringe parameter measurement, we are actually examining the limitations to the measurability of coherence itself. In this paper, fundamental limitations of estimating the amplitudes and phases of polar-interferograms recorded at low light levels are investigated. By modeling the receiver as a spatial array of photon-counting detectors, results are obtained that permit specification of the minimum number of photoevents required for estimation of fringe parameters to a given accuracy. Both a discrete Fourier-transform estimator and an optimum joint maximum-likelihood estimator are considered to specify the limiting performance of all unbiased estimators in terms of the collected light flux.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1]–[3]. Our experimental setup (see Fig. 1 (left)) allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1–3]. Our experimental setup allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].
The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state
∣ψ⟩. Having access to
m copies of an unknown pure state
∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success
≥2/3 is
Θ(c(∣ψ⟩)−1), where
c(∣ψ⟩) is the geometric measure of coherence of
∣ψ⟩. However, assuming access to the unitary
Uψ which prepares the unknown state
∣ψ⟩, and its inverse
Uψ†, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number
O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error
ε with a sample complexity that scales as
O(1/ε) as compared to the
O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
Interferometric methods form the basis of highly sensitive measurement techniques from astronomy to bioimaging. Interferometry typically requires high stability between the measured and reference beams. The presence of rapid phase fluctuations washes out interference fringes, making phase profile recovery impossible. This challenge can be addressed by shortening the measurement time. However, such an approach reduces photon-counting rates, precluding applications in low-intensity imaging. We introduce a phase imaging technique which is immune to time-dependent phase fluctuations. Our technique, relying on intensity correlation instead of direct intensity measurements, allows one to obtain high interference visibility for arbitrarily long acquisition times. We prove the optimality of our method using the Cramér-Rao bound in the extreme case when no more than two photons are detected within the time window of phase stability. Our technique will broaden prospects in phase measurements, including emerging applications such as in infrared and x-ray imaging and quantum and matter-wave interferometry.