Abstract Physical realizations of the canonical phase measurement for the optical phase are unknown. Single-shot phase estimation, which aims to determine the phase of an optical field in a single shot, is critical in quantum information processing and metrology. Here we present a family of strategies for single-shot phase estimation of coherent states based on adaptive non-Gaussian, photon counting, measurements with coherent displacements that maximize information gain as the measurement progresses, which have higher sensitivities over the best known adaptive Gaussian strategies. To gain understanding about their fundamental characteristics and demonstrate their superior performance, we develop a comprehensive statistical analysis based on Bayesian optimal design of experiments, which provides a natural description of these non-Gaussian strategies. This mathematical framework, together with numerical analysis and Monte Carlo methods, allows us to determine the asymptotic limits in sensitivity of strategies based on photon counting designed to maximize information gain, which up to now had been a challenging problem. Moreover, we show that these non-Gaussian phase estimation strategies have the same functional form as the canonical phase measurement in the asymptotic limit differing only by a scaling factor, thus providing the highest sensitivity among physically-realizable measurements for single-shot phase estimation of coherent states known to date. This work shines light into the potential of optimized non-Gaussian measurements based on photon counting for optical quantum metrology and phase estimation.
Abstract Physical realizations of the canonical phase measurement for the optical phase are unknown. Single-shot phase estimation, which aims to determine the phase of an optical field in a single shot, is critical in quantum information processing and metrology. Here we present a family of strategies for single-shot phase estimation of coherent states based on adaptive non-Gaussian, photon counting, measurements with coherent displacements that maximize information gain as the measurement progresses, which have higher sensitivities over the best known adaptive Gaussian strategies. To gain understanding about their fundamental characteristics and demonstrate their superior performance, we develop a comprehensive statistical analysis based on Bayesian optimal design of experiments, which provides a natural description of these non-Gaussian strategies. This mathematical framework, together with numerical analysis and Monte Carlo methods, allows us to determine the asymptotic limits in sensitivity of strategies based on photon counting designed to maximize information gain, which up to now had been a challenging problem. Moreover, we show that these non-Gaussian phase estimation strategies have the same functional form as the canonical phase measurement in the asymptotic limit differing only by a scaling factor, thus providing the highest sensitivity among physically-realizable measurements for single-shot phase estimation of coherent states known to date. This work shines light into the potential of optimized non-Gaussian measurements based on photon counting for optical quantum metrology and phase estimation.
Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.
Phase estimation plays a central role in communications, sensing, and information processing. Quantum correlated states, such as squeezed states, enable phase estimation beyond the shot-noise limit, and in principle approach the ultimate quantum limit in precision, when paired with optimal quantum measurements. However, physical realizations of optimal quantum measurements for optical phase estimation with quantum-correlated states are still unknown. Here we address this problem by introducing an adaptive Gaussian measurement strategy for optical phase estimation with squeezed vacuum states that, by construction, approaches the quantum limit in precision. This strategy builds from a comprehensive set of locally optimal POVMs through rotations and homodyne measurements and uses the Adaptive Quantum State Estimation framework for optimizing the adaptive measurement process, which, under certain regularity conditions, guarantees asymptotic optimality for this quantum parameter estimation problem. As a result, the adaptive phase estimation strategy based on locally-optimal homodyne measurements achieves the quantum limit within the phase interval of [ 0 , π / 2 ) . Furthermore, we generalize this strategy by including heterodyne measurements, enabling phase estimation across the full range of phases from [ 0 , π ) , where squeezed vacuum allows for unambiguous phase encoding. Remarkably, for this phase interval, which is the maximum range of phases that can be encoded in squeezed vacuum, this estimation strategy maintains an asymptotic quantum-optimal performance, representing a significant advancement in quantum metrology.
Using multi-photon entangled input states, we estimate the phase uncertainty in a noiseless Mach–Zehnder interferometer using photon-counting detection. We assume a flat prior uncertainty and use Bayesian inference to construct a posterior uncertainty. By minimizing the posterior variance to get the optimal input states, we first devise an estimation and measurement strategy that yields the lowest phase uncertainty for a single measurement. N00N and Gaussian states are determined to be optimal in certain regimes. We then generalize to a sequence of repeated measurements, using non-adaptive and fully adaptive measurements. N00N and Gaussian input states are close to optimal in these cases as well, and optimal analytical formulae are developed. Using these formulae as inputs, a general scaling formula is obtained, which shows how many shots it would take on average to reduce phase uncertainty to a target level. Finally, these theoretical results are compared with a Monte Carlo simulation using frequentist inference. In both methods of inference, the local non-adaptive method is shown to be the most effective practical method to reduce phase uncertainty.
Phase estimation is known to be a robust method for single-qubit gate calibration in quantum computers, while Bayesian estimation is widely used in devising optimal methods for learning in quantum systems. We present a Bayesian method that adaptively chooses a control phase and the time of coherent evolution. The uniqueness of the approach is to use analytics to implement an efficient numerical algorithm that offers exact calculation of the Bayesian probability and associated utility functions, even in the presence of noise. We determine the utility of control parameter values using functions of the prior probability of the phase that quantify expected knowledge gain in terms of either expected narrowing of the posterior or expected information gain. We find that by maximizing the of expected gain we obtain phase estimates having standard deviation a factor of 1.43 larger than the Heisenberg limit using a classical sequential strategy for noise-free experiments. In the presence of dephasing, we find near-optimal performance with respect to the Cramér-Rao bound. We also demonstrate some robustness of the estimates to noise that is not accounted for in the model of the estimator, making the methods suitable for calibrating operations in quantum computers. The rate of gain can be adjusted to account for times in the experiment other than the coherent evolution of the unknown phase, such as times required for state preparation or readout. The combined features of the method make it well suited to single-qubit gate calibration in trapped-ion or neutral atom experiments and promises a wide range of other applications by offering near-optimal phase estimation that straightforwardly accounts for experimental imperfections.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
Summary form only given. The use of quantum states of light in optical interferometry improves the precision in the estimation of a phase shift, paving the way for applications in quantum metrology, computation and cryptography. Sub-shot noise phase sensing can for example be achieved by injecting a squeezed vacuum into an interferometer . However, this approach leads to enhanced sensitivity only for small phase shifts. In this work we aim for ab initio sub-shot noise estimation of an unknown phase shift using a pre-determined squeezed probe and an adaptive measurement approach. We experimentally investigate the performances of such protocol under the realistic assumption of thermalization of the probe state. Indeed, adaptive phase estimation schemes with squeezed states and Bayesian processing of homodyne data have been shown to be asymptotically optimal in the pure case, thus approaching the quantum Cramér-Rao bound. In our protocol we take advantage of the enhanced sensitivity of homodyne detection in proximity of the optimal phase which maximizes the homodyne Fisher information. A squeezed thermal probe state (signal) undergoes an unknown phase shift. The first estimation step involves interference on a beam splitter of the signal and a local oscillator followed by homodyne detection. Homodyne data is then processed to compute a rough estimation of the phase through Bayesian inference. The rough estimation is fed back to the local oscillator in order to match the optimal relative phase with the signal. A second estimation step leads to the final estimation of the phase shift. Thermalization of the probe state prevents the attainability of the quantum Cramér-Rao bound. Nevertheless, we show that the studied adaptive scheme still saturates the classical Cramér-Rao bound, showing sub-shot noise behaviour and therefore extracting the maximum available information from homodyne data. In contrast to previous approaches, our scheme is optimized for Gaussian states.