Optimal phase space projection for noise reduction

In this communication we will re-examine the widely studied technique of phase space projection. By imposing a time domain constraint (TDC) on the residual noise, we deduce a more general version of the optimal projector, which includes those appearing in previous literature as subcases but does not assume the independence between the clean signal and the noise. As an application, we will apply this technique for noise reduction. Numerical results show that our algorithm has succeeded in augmenting the signal-to-noise ratio (SNR) for simulated data from the R\"ossler system and experimental speech record.Comment: Accepted version for PR

Bayesian estimation of one-parameter qubit gates

We address estimation of one-parameter unitary gates for qubit systems and seek for optimal probes and measurements. Single- and two-qubit probes are analyzed in details focusing on precision and stability of the estimation procedure. Bayesian inference is employed and compared with the ultimate quantum limits to precision, taking into account the biased nature of Bayes estimator in the non asymptotic regime. Besides, through the evaluation of the asymptotic a posteriori distribution for the gate parameter and the comparison with the results of Monte Carlo simulated experiments, we show that asymptotic optimality of Bayes estimator is actually achieved after a limited number of runs. The robustness of the estimation procedure against fluctuations of the measurement settings is investigated and the use of entanglement to improve the overall stability of the estimation scheme is also analyzed in some details.Comment: 10 pages, 5 figure

Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing

Classical and quantum theories of time-symmetric smoothing, which can be used to optimally estimate waveforms in classical and quantum systems, are derived using a discrete-time approach, and the similarities between the two theories are emphasized. Application of the quantum theory to homodyne phase-locked loop design for phase estimation with narrowband squeezed optical beams is studied. The relation between the proposed theory and Aharonov et al.'s weak value theory is also explored.Comment: 13 pages, 5 figures, v2: changed the title to a more descriptive one, corrected a minor mistake in Sec. IV, accepted by Physical Review

Fundamental quantum limits to waveform detection

Ever since the inception of gravitational-wave detectors, limits imposed by quantum mechanics to the detection of time-varying signals have been a subject of intense research and debate. Drawing insights from quantum information theory, quantum detection theory, and quantum measurement theory, here we prove lower error bounds for waveform detection via a quantum system, settling the long-standing problem. In the case of optomechanical force detection, we derive analytic expressions for the bounds in some cases of interest and discuss how the limits can be approached using quantum control techniques.Comment: v1: first draft, 5 pages; v2: updated and extended, 5 pages + appendices, 2 figures; v3: 8 pages and 3 figure

Fisher Information for Inverse Problems and Trace Class Operators

This paper provides a mathematical framework for Fisher information analysis for inverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriate space for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictly smaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information to be trace class, a sufficient condition is formulated based on the singular values of the Jacobian as well as of the eigenvalues of the covariance operator, together with some regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with "external" spherically isotropic noise, as well as "internal" additive uncorrelated noise.Comment: Submitted to Journal of Mathematical Physic

Quantum theory of optical temporal phase and instantaneous frequency

We propose a general quantum theory of optical phase and instantaneous frequency in the time domain for slowly varying optical signals. Guided by classical estimation theory, we design homodyne phase-locked loops that enable quantum-limited measurements of temporal phase and instantaneous frequency. Standard and Heisenberg quantum limits to such measurements are then derived. For optical sensing applications, we propose multipass and Fabry-P\'erot position and velocity sensors that take advantage of the signal-to-noise-ratio enhancement effect of wideband angle modulation without requiring nonclassical light. We also generalize our theory to three spatial dimensions for nonrelativistic bosons and define an Hermitian fluid velocity operator, which provides a theoretical underpinning to the current-algebra approach of quantum hydrodynamics.Comment: 16 pages, v3: rewritten and extended, v4: some minor mistakes corrected, accepted by Physical Review