In this thesis we make progress towards applications of quantum estimation theory to new physical systems. We first consider two commonly visited problems in quantum metrology: source optimisation and source localisation. For the first, we focus on estimating the distances, d, between neighbouring light sources along an array, which undergoes stretching deformations. We evaluate how changing the nature of the sources impacts the estimation precision of d by using the quantum Fisher information (QFI) as a figure of merit. By comparing this quantity for arrays of single photon emitters, coherent, thermal, and entangled sources, we find that the classical coherent and thermal states outperform the single photon emitters. This would be favourable since generating classical states is less resource-expensive to create. However, a quantum enhancement is observed when entanglement is employed. In agreement with separate work, the optimal state is that which entangles the eigenstates corresponding to the maximum and minimum difference eigenvalues of the generator. We demonstrate that not all entangled states can reproduce similar precision enhancements. This insight is reminiscent of previous studies, where entanglement was concluded as a necessary but insufficient resource for quantum metrology.
Next, we address the source localisation problem to detect any deformations applied to a grid of sources. Improving this detection depends on our ability to engineer grids that maximise the sensitivity of the QFI matrix. Hence, we derive the generators of local translations of unitary evolutions that describe any general grid deformation, and show that our result is a multi-parameter extension of other results in the literature. We obtain a general result for the quantum Fisher information matrix (QFIM) through these generators for any grid deformation and explore specific spatial maps, including composite stretches, shears, and rotations. Since the QFI matrix depends only on the properties of the probe state and the configuration of the emitters, we explore how we can modify both to enhance our estimation sensitivity to determine the applied grid deformation. Physically motivated, we find the best arrangement of sources that enhances the sensitivity of detection for a set number of sources.
Finally, we consider the optimal estimation of a complex squeezing operation in phase space. The use of squeezed light as a quantum resource is ubiquitous in quantum optics, and a complete characterisation of a complex squeezing operation is pivotal for fundamental reasons. This is a true multi-parameter quantum estimation problem of incompatible observables. Specifically, we find that the symmetric logarithmic derivates (SLDs) for amplitude and directional squeezing do not commute. This prohibits simultaneous optimal estimates of both parameters, even in the asymptotic limit. As a result, we focus on finding separable optimal estimates. The Cramer-Rao bound is determined to provide a theoretical benchmark on the bi-variate estimation precision for general single mode Gaussian probes. Using this and the SLDs, we present a practical experimental implementation that can realise the individual fundamental precision bounds.
Multimode Gaussian quantum light, which includes multimode squeezed and multipartite quadrature entangled light, is a very general and powerful quantum resource with promising applications in quantum information processing and metrology. In this paper, we determine the ultimate sensitivity in the estimation of any parameter when the information about this parameter is encoded in such light, irrespective of the information extraction protocol used in the estimation and of the quantity measured. In addition we show that an appropriate homodyne detection scheme allows us to reach the Quantum Cramer-Rao bound. We show that, for a given set of available quantum resources, the most economical way to maximize the sensitivity is to put the most squeezed state available in a well-defined light mode. This implies that it is not relevant to take advantage of the existence of squeezed fluctuations in other modes, nor of quantum correlations and entanglement between different modes. We finally apply these considerations to the problem of optimal phase shift estimation.
We introduce spatial deformations to an array of light sources and study how the estimation precision of the interspacing distance
d changes with the sources of light used. The quantum Fisher information (QFI) is used as the figure of merit in this work to quantify the amount of information we have on the estimation parameter. We derive the generator of translations $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{G}}$ in
d due to an arbitrary homogeneous deformation applied to the array. We show how the variance of the generator can be used to easily consider how different deformations and light sources can effect the estimation precision. The single-parameter estimation problem is applied to the array, and we report on the optimal state that maximizes the QFI for
d. Contrary to what may have been expected, the higher average mode occupancies of the classical states performs better in estimating
d when compared with single photon emitters (SPEs). The optimal entangled state is constructed from the eigenvectors of the generator and found to outperform all these states. We also find the existence of multiple optimal estimators for the measurement of
d. Our results find applications in evaluating stresses and strains, fracture prevention in materials expressing great sensitivities to deformations, and selecting frequency distinguished quantum sources from an array of reference sources.