periodicity of phase readout introduces a wrap-around ambiguity, which confines the unambiguous measurement range to a single period and leads to a structural trade-off between precision and measurement range in high-sensitivity settings. To address this problem, we propose a robust Chinese remainder theorem-based multi-baseline measurement scheme: we choose a set of pairwise coprime scale factors and recover the wrap-around number from congruence relations among the reduced phases. In the setting of optical-interferometric quantum telescopes, we construct a telescope array based on the Chinese remainder theorem using fractional baselines, which substantially enlarges the unambiguous angular range (effective field of view) while retaining the high precision enabled by long baselines. Numerical simulations show that, under the same constraint on the shortest achievable baseline, our scheme achieves an exponential improvement in detection range over existing methods and attains a better range-precision trade-off with fewer probing resources. This framework supports practical multi-baseline quantum-interferometric observations and can be generalized to other phase-sensitive quantum precision-measurement tasks.
periodicity of phase readout introduces a wrap-around ambiguity, which confines the unambiguous measurement range to a single period and leads to a structural trade-off between precision and measurement range in high-sensitivity settings. To address this problem, we propose a robust Chinese remainder theorem-based multi-baseline measurement scheme: we choose a set of pairwise coprime scale factors and recover the wrap-around number from congruence relations among the reduced phases. In the setting of optical-interferometric quantum telescopes, we construct a telescope array based on the Chinese remainder theorem using fractional baselines, which substantially enlarges the unambiguous angular range (effective field of view) while retaining the high precision enabled by long baselines. Numerical simulations show that, under the same constraint on the shortest achievable baseline, our scheme achieves an exponential improvement in detection range over existing methods and attains a better range-precision trade-off with fewer probing resources. This framework supports practical multi-baseline quantum-interferometric observations and can be generalized to other phase-sensitive quantum precision-measurement tasks.
The Chinese remainder theorem is used in metrology for extending the range of quantum clocks/radar/interferometry, where the phase of a signal is known relative to a set of oscillators with different periods. This paper investigates the performance of a quantum-mechanical Chinese remainder clock, consisting of atoms/oscillators with pairwise coprime periods. We provide the optimal initial state and the optimal Heisenberg-limited quantum measurements for measuring time up to the product of the periods. We introduce a novel fault-tolerant post-processing protocol that allows reconstruction of the correct time even in the presence of errors in the remainders.
Abstract Precise angular estimation of remote targets is a fundamental task in optical interferometry, particularly in the weak-light regime where photon resources are limited. Interferometric schemes employing multiple telescopes can overcome the diffraction limit of a single aperture but are often highly sensitive to experimental imperfections. In this work, we study a piecemeal interferometric approach that jointly exploits measurement data from multiple baselines for angular estimation, and we relax the constraints on baseline configurations. The proposed scheme remains effective even with a reduced number of baselines, highlighting its practical applicability. Focusing on realistic experimental conditions, we analyze the impact of baseline length and orientation errors on the estimation performance. We show that both the original and generalized piecemeal protocols exhibit strong robustness against these imperfections and achieve higher angular estimation precision than existing single-baseline weak-light interferometric schemes under identical practical conditions.
Closure phase enables imaging with ground-based interferometric arrays by canceling telescope-dependent phase errors due to atmospheric fluctuations. It can be obtained by measuring fringe phases on individual baselines forming a triangle and summing them. We show that closure phases can be measured more accurately with fewer photons by performing joint measurements on multiple photons. We first establish that at least three photons from the same atmospheric temporal phase-stability window are required for closure-phase information to survive the unknown telescope-dependent phase noise, and show that joint three-photon measurements already provide an advantage over single-photon measurements in the minimal three-telescope setting. We then turn to large arrays and consider estimating all bispectra---complex quantities whose phases are the closure phases---under a global mean-square error criterion. Although an m-telescope array contains Θ(m2) independent triangle bispectra, we show that physical constraints on the visibilities make the total information content grow only nearly linearly with m at fixed accuracy. Exploiting this structure, we construct an efficient protocol using at most three-photon joint measurements with photon cost O~(m), whereas any sequential single-photon protocol requires Ω(m2) photons. When the number of detected photons per temporal stability window scales linearly with the array size, this allows the required exposure time to remain independent of m up to logarithmic factors. We further analyze the complementary high-precision regime using multiparameter quantum metrology and find a similar advantage for collective measurements. Overall, our results show that quantum information processing techniques can substantially improve the sensitivity and robustness of optical interferometry.
In precision measurement and quantum metrology, the $2\ensuremath{\pi}$ periodicity inherent to any interferometric signal sets a fundamental limit in the simultaneous achievement of both high measurement sensitivity and range. Here, based on the extended period of the Lissajous curve, we develop a method to break this sensitivity--maximum-range limit completely. In the measurement of magnetic field using an electronic spin in diamond under ambient conditions, we extend the maximum range by 2 orders of magnitude without sacrificing detection sensitivity and bandwidth for sufficiently long measurement time. Furthermore, we achieve precise detection of a time-varying periodic waveform, where a large measurement range can induce high accuracy and demonstrate a Heisenberg-like scaling within the coherent regime. This Lissajous-curve-based range extension method can be directly applied to any other modulo-limited quantum interferometers, which makes a useful and indispensable stepping stone toward practical interferometric quantum metrology.
A quantum-enhanced telescope (interferometer) can have significant advantages in precisely measuring the angular position of remote objects through sending out ancillary quantum entanglement pairs alongside the large baselines of telescopes. In principle, the larger the baseline, the higher the expected measurement resolution. However, the well-known major problem of photon transmission loss would drastically undermine the practical application of all quantum-enhanced telescopes and optical interferometers that transmit photons along the baseline, especially for the large baselines required for high angular resolution. Here, we propose a photon-transmission-loss-independent quantum-enhanced interferometer that is robust to the transmission loss and keeps its advantages unchanged given whatever large baseline. This removes the well-known barrier to the practical application of quantum-enhanced telescopes. Moreover, in contrast to the previous results of quantum-entanglement-enhanced telescopes requesting sophisticated resources—perfect single-photon entanglement, photon-number-resolving detectors, or quantum memories—our method requires only threshold detectors and tunable coherent-state or two-mode squeezed-state sources, which are mature technologies nowadays, and it does not need any quantum memory. We demonstrate the high performance of our method using Fisher information. For the phase difference of weak light from a remote star, the high-quality result of Fisher information remains constant under arbitrarily large transmission loss, and hence all the advantageous results and the performance of our method are actually independent of transmission loss.
Telescope systems with distributed apertures are a well-established approach for boosting resolution in astronomical imaging. However, theoretical limits on quantitative imaging precision, and the fundamentally best possible mode-sorting, beam-combining, and detection schemes to use with such arrays, remain largely unexplored. Using mathematical tools of the quantum and classical Crámer-Rao bounds, we perform analyses showing the fundamental origins of the enhancement provided by distributed imaging systems, over and above a single monolithic telescope, and consider the precision with which one can estimate any desired parameter embedded in a scene's incoherent radiation with a multi-aperture imaging system. We show how quantum-optimal measurements can be realized via beam-combination strategies of two classes: (1) , where light from different apertures is directed to a common focal plane, e.g., of a segmented-aperture telescope; and (2) , where light collected at each aperture, e.g., telescope sites of a long-baseline array, is routed to an optical interferometer. As an example, we show an explicit calculation of the quantum Fisher information (QFI) for estimating the angular separation between two-point emitters using two identical apertures separated by a baseline distance. We show that this QFI splits instructively into additive contributions from the single apertures and from the baseline. We quantify the relative benefits of intratelescope (e.g., spatial-mode) optical processing and intertelescope beam combination. We show how both receiver designs can be used to capture both sources of information and discuss how similar methods could be extended to more general imaging tasks. We discuss translating QFI-attaining measurements to explicit receiver designs, and the use of preshared entanglement to achieve the QFI when it is impractical to colocate and combine light collected by the apertures. Published by the American Physical Society 2024