Displacement based quantum receivers can beat the standard quantum limit and have gained increasing attention in recent years. The displacement operation value can be estimated and calibrated by using Fock states as probes. In this Letter, we focus on the impacts of thermal noise when preparing the Fock probes on the estimation performance. We first derive the density matrix and the photon statistics of the noisy displaced number state with a uniformly distributed displacement phase. Based on the derived density matrix, we then analyze the performance of estimating displacement operations by using the maximum likelihood estimation (MLE) method. To decrease the computational complexity of the MLE, we further derive an approximated probability density of the photon numbers under small displacement. Besides, the quantum Cramér–Rao bound (QCRB) is also derived to analyze the performance limit of the displacement estimation. Numerical results found that, though thermal noise can greatly degrade the MLE accuracy of the displacement operations, the performance advantage of using Fock probes still persists over classical coherent probes. Besides, we demonstrate that the MLE method based on Fock probes can well approach the QCRB when the displacement strength N c ≤ ~1.
Displacement based quantum receivers can beat the standard quantum limit and have gained increasing attention in recent years. The displacement operation value can be estimated and calibrated by using Fock states as probes. In this Letter, we focus on the impacts of thermal noise when preparing the Fock probes on the estimation performance. We first derive the density matrix and the photon statistics of the noisy displaced number state with a uniformly distributed displacement phase. Based on the derived density matrix, we then analyze the performance of estimating displacement operations by using the maximum likelihood estimation (MLE) method. To decrease the computational complexity of the MLE, we further derive an approximated probability density of the photon numbers under small displacement. Besides, the quantum Cramér–Rao bound (QCRB) is also derived to analyze the performance limit of the displacement estimation. Numerical results found that, though thermal noise can greatly degrade the MLE accuracy of the displacement operations, the performance advantage of using Fock probes still persists over classical coherent probes. Besides, we demonstrate that the MLE method based on Fock probes can well approach the QCRB when the displacement strength N c ≤ ~1.
Displacement based quantum receivers have gained increasing attention because they can beat the standard quantum limit. The displacement value in quantum receivers can be calibrated by using number states as probes. Current estimating methods assumed that ideal number states can be prepared perfectly, which is impossible in practical implementations. In this work, we focus on the impacts of thermal noise when preparing the number states on the estimation performance of displacement operations. We first derive the probability density of photon numbers in a thermal noise contaminated number state after a displacement operation, which we call the noisy displaced number state (NDNS). Based on the derived probability density, we then analyze the performance of estimating displacement operations with NDNS probes by using the maximum likelihood estimation (MLE) method. Numerical results found that the thermal noise in preparing the number states has great influence on the MLE accuracy of the displacement operations.
Abstract Since a general Gaussian process is phase-sensitive, a stable phase reference is required to take advantage of this feature. When the reference is missing, either due to the volatile nature of the measured sample or the measurement’s technical limitations, the resulting process appears as random in phase. Under this condition, we consider two single-mode Gaussian processes, displacement and squeezing. We show that these two can be efficiently estimated using photon number states and photon number resolving detectors. For separate estimation of displacement and squeezing, the practical estimation errors for hundreds of probes’ ensembles can saturate the Cramér–Rao bound even for arbitrary small values of the estimated parameters and under realistic losses. The estimation of displacement with Fock states always outperforms estimation using Gaussian states with equivalent energy and optimal measurement. For estimation of squeezing, Fock states outperform Gaussian methods, but only when their energy is large enough. Finally, we show that Fock states can also be used to estimate the displacement and the squeezing simultaneously.
The performance of Kennedy receiver with threshold detection is analyzed in the presence of thermal noise. A lower bound is first derived for the error probability of the Kennedy receiver in the presence of thermal noise and it is shown that the performance of the Kennedy receiver is greatly affected by the thermal noise. Then we analyze the performance of the Kennedy receiver using the maximum a posteriori detection, which is proven to be equivalent to a threshold detection. We also optimize the displacement operation to improve further the receiver performance. Numerical results show that the Kennedy receiver with threshold detection can surpass the standard quantum limit achieved by homodyne detection when the average number of thermal photons is smaller than 0.1. The Kennedy receiver with threshold detection using optimized displacement can surpass the standard quantum limit when the average number of thermal photons is smaller than 0.2.
We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed [D. Gottesman et al., Phys. Rev. A 64, 012310 (2001)] for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed [B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016)]. We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cram\'er-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities, and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.
We present a method to estimate the amount of displacement, squeezing and temperature of a single-mode harmonic oscillator state based on both the weighted least squares and maximum likelihood estimators applied to measured Fock state populations and fluorescence measurements. Displacement, squeezing and temperature, or equivalently the mean of the quadratures and its variances, are essential state parameters used in quantum computation and various communication and sensing protocols. They are often measured with homodyne-style detection, which requires a phase reference such as a local oscillator. Our method allows estimation without a phase reference, by using for example a photon-number-resolving detector. To evaluate the performance of our estimator, we simulated experiments with different values of displacement, squeezing, phase and temperature. First, we simulate Fock population measurement experiments. From 10,000 Fock measurement events we produced estimates for states whose fidelities to the true state are greater than 99.99% for small squeezing (r < 1.0), and for high squeezing (r = 2.5) we obtain fidelities greater than 99.9%. We also report confidence intervals and their coverage probabilities, mean squared error and their coverage probabilities. We also simulate fluorescence measurement experiments for different motional states of a trapped ion. Jointly fitting datasets for different relative phases helped to decrease significantly the uncertainty on the estimates. Based on the analysis of real data we proposed a correction to the model that describes the probability of measuring the ion on the spin down state.
For a given quantum state used in sensing, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the precision achievable by an unbiased estimator of an unknown parameter, determined by the inverse of the quantum Fisher information (QFI). The QFI serves as an upper bound on the classical Fisher information (CFI), representing the maximum extractable information about the unknown parameter from measurements on a physical system. Thus, a central goal in quantum parameter estimation is to find a measurement, described by a POVM, that saturates the QFI (achieves maximum CFI), and thereby achieves the QCRB. In the idealization that one uses pure states and unitary encodings for sensing, discontinuities can appear in the CFI but not the QFI. In this article, we demonstrate that these discontinuities are important features, quantifying how much Fisher information is lost in the presence of noise. We refer to this as the Fisher information "fragility". We present a simple framework for understanding how discontinuities increase fragility through Jensen's inequality, and demonstrate how one can use this framework to design more robust POVMs for quantum advantage in metrology.