Our knowledge about surroundings can be achieved by observations and measurements but both are influenced by errors (noise). Therefore one of the first tasks is to try to eliminate the noise by constructing instruments with high accuracy. But any real observed and measured system is characterized by natural limits due to the deterministic nature of the measured information. The present work is dedicated to the identification of these limits. We have analyzed some algorithms for selection and estimation based on statistical hypothesis and we have developed a theoretical method for their validation. A classic (non-quantic) algorithm for observations and measurements based on statistical strategies of optical field is presented in detail. A generalized statistical strategy for observations and measurements on the nuclear particles, is based on these results, taking into account the particular type of statistics resulting from the measuring process also.
Our knowledge about surroundings can be achieved by observations and measurements but both are influenced by errors (noise). Therefore one of the first tasks is to try to eliminate the noise by constructing instruments with high accuracy. But any real observed and measured system is characterized by natural limits due to the deterministic nature of the measured information. The present work is dedicated to the identification of these limits. We have analyzed some algorithms for selection and estimation based on statistical hypothesis and we have developed a theoretical method for their validation. A classic (non-quantic) algorithm for observations and measurements based on statistical strategies of optical field is presented in detail. A generalized statistical strategy for observations and measurements on the nuclear particles, is based on these results, taking into account the particular type of statistics resulting from the measuring process also.
A mode of quantic detection for noncoherent optical radiation as noise limit is presented. Using the representation on coherent state of density matrix operators ((rho) ) and of detection operators ((pi) ), the detection probability (Qd), false alarm probability (Qo) and signal-noise ratio (S/Z), are estimated, as function of the number of freedom degrees of temporal oscillation modes (Mt) and of the spatial oscillation modes (Ms). A validation algorithm of statistical hypotheses, resulted from noncoherent optical field analysis, is made using Newman-Pearson criterion.
This thesis reports the development of a quantum enhanced microparticle tracking technique, which applied to measurements within living cells, allowed the first demonstrations of both biological measurements beyond the quantum noise limit and quantum enhanced spatial resolution within biology. The sensitivity of any optical measurement is limited by noise which follows from quantization of the electromagnetic field. This thesis theoretically characterizes the resulting quantum limit on particle tracking precision; both in the limit of perfect measurements and for the measurements used in real experiments, and theoretically establishes an experimental strategy which could allow the quantum limit to be surpassed via the application of non-classical light. Further, a computational tool is developed which allows rapid characterization of particle tracking experiments, thus providing researchers the benefit of rigorous theoretical predictions without requiring detailed calculations. Following this, classical technologies are developed which can improve microparticle tracking sensitivity and which are enabling steps toward integration of non-classical light. An interferometric strategy is described whereby signal-to-noise is improved by removing unwanted light from the detector. An optical lock-in technique is developed which eliminates low-frequency noise, thus allowing quantum noise limited performance at low frequencies. This is a crucial requirement for quantum enhanced measurements, and can also improve precision in classical experiments. Dark-field illumination is also explored as a method to remove unwanted background light, and the optimal illumination angle is calculated for our intended experiments. These advances are then applied in the development of the first quantum enhanced particle tracking apparatus. Squeezed states of light are used to improve particle tracking precision by 2.7 dB, demonstrating quantum enhanced microscopy for the first time. This was applied to perform the first measurements of living systems with quantum enhanced precision. Naturally occurring lipid particles were tracked within the cytoplasm of Saccharomyces cerevisiae yeast cells, with squeezed light enhancing precision by 2.4 dB. The thermal motion of these particles could then be analyzed to infer the viscosity and elasticity of the local environment, with squeezed light allowing 64% faster observations, thus improving the sensitivity to dynamic biological changes. This experiment was then extended to spatially resolved quantum imaging, with lipid particles used as scanning probes in a technique called photonic force microscopy. The particles sample variations in the local environment with resolution limited by measurement precision rather than the diffraction limit. The use of squeezed light was found to enhance spatial resolution by 14% at an absolute resolution level of 10 nm, comparable to leading classical experiments. This was the first demonstration of both sub-diffraction limited resolution and quantum enhanced resolution in biology, and places practical applications of quantum imaging within reach. Finally, the future challenges and prospects of quantum enhanced particle tracking are outlined.n
The study of the wave phenomena of physical optics and of quantum mechanics must take into account the fact that experimental instruments operate linearly on the field quantities, while all measured values depend only on their mean squares. The latter circumstance necessitates a statistical approach to these theories. Certain questions must be re-examined in the light of this fact. We analyze the most important of these, namely that of the limitations causality (the directedness of time in physical phenomena) imposes on the nature of the transformations an optical instrument can perform on the light admitted to it.
It is suggested that computers exploiting quantum phenomena can solve tasks with an efficiency surpassing that achievable by classical computers. One example of this quantum advantage arises in the task of phase estimation. Here, an unknown eigenvalue of a unitary evolution is estimated, a critical subroutine for many quantum algorithms and quantum metrology protocols. For a fixed amount of resources, the utilisation of quantum coherence and entanglement can lead to a quadratic increase in precision of the final estimate compared to classical statistical limits. In systems affected by depolarising noise, including current quantum hardware, this improvement becomes a constant factor. While numerous phase estimation algorithms have been proposed claiming to achieve these theoretical limits, the lack of well-defined criteria for both precision and resource utilisation casts doubt on these results. Further issues arise due to a lack of point identification (the existence of multiple equally likely estimates) that some algorithms have. I aim to address these issues in this thesis. I employ statistical inference, the theoretical foundation for interpreting any quantum measurement, in order to find rigorous definitions for precision and resource use. By combining these definitions with previous work in the field of quantum metrology, I am able to derive theoretical precision limits that one can hope to achieve. I then outline several phase estimation algorithms that can achieve these limits in different contexts. These algorithms that I describe are split into three major groups. Algorithms in the first group focus on the process of point identification; one set of measurements provides a precise estimate of the unknown phase before an additional set of measurements removes any ambiguities present. This second set of measurements can be either decided before any measurement is taken or during measurements in an iterative fashion. I show that the balancing of resources used in each set of measurements can lead to optimally scaling precision both with and without depolarising noise. These algorithms are based upon two of my peer-reviewed articles, [Phys. Rev. A 106, 062615 (2022)] and [Phys. Rev. A 109, 042412 (2024)]. The second group of algorithms that I introduce use the principles of Bayesian experimental design. Here, previous measurement outcomes are used to select future measurements recursively. These algorithms require large classical computing times to select each measurement but boast high precision in certain metrics. The final group of algorithms exploit the oscillating nature of phases by using a discrete Fourier transform (DFT). The DFT can be implemented using either classical or quantum hardware, the latter of which leads to the famous quantum phase estimation algorithm (QPEA). In this thesis, I show that the QPEA performs worse than some classical precision limits. However, I also suggest some modifications to the QPEA circuit that can lead to optimal precision scaling. These results and modifications are described in one of my articles awaiting publication [arXiv:2505.00764].
The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.