Quantum metrology protocols are typically designed around the assumption that we have an abundance of measurement data, but recent practical applications are increasingly driving interest in cases with very limited data. In this regime the best approach involves an interesting interplay between the amount of data and the prior information. Here we propose a new way of optimising these schemes based on the practically-motivated assumption that we have a sequence of identical and independent measurements. For a given probe state we take our measurement to be the best one for a single shot and we use this sequentially to study the performance of different practical states in a Mach-Zehnder interferometer when we have moderate prior knowledge of the underlying parameter. We find that we recover the quantum Cramér-Rao bound asymptotically, but for low data counts we find a completely different structure. Despite the fact that intra-mode correlations are known to be the key to increasing the asymptotic precision, we find evidence that these could be detrimental in the low data regime and that entanglement between the paths of the interferometer may play a more important role. Finally, we analyse how close realistic measurements can get to the bound and find that measuring quadratures can improve upon counting photons, though both strategies converge asymptotically. These results may prove to be important in the development of quantum enhanced metrology applications where practical considerations mean that we are limited to a small number of trials.
Quantum metrology protocols are typically designed around the assumption that we have an abundance of measurement data, but recent practical applications are increasingly driving interest in cases with very limited data. In this regime the best approach involves an interesting interplay between the amount of data and the prior information. Here we propose a new way of optimising these schemes based on the practically-motivated assumption that we have a sequence of identical and independent measurements. For a given probe state we take our measurement to be the best one for a single shot and we use this sequentially to study the performance of different practical states in a Mach-Zehnder interferometer when we have moderate prior knowledge of the underlying parameter. We find that we recover the quantum Cramér-Rao bound asymptotically, but for low data counts we find a completely different structure. Despite the fact that intra-mode correlations are known to be the key to increasing the asymptotic precision, we find evidence that these could be detrimental in the low data regime and that entanglement between the paths of the interferometer may play a more important role. Finally, we analyse how close realistic measurements can get to the bound and find that measuring quadratures can improve upon counting photons, though both strategies converge asymptotically. These results may prove to be important in the development of quantum enhanced metrology applications where practical considerations mean that we are limited to a small number of trials.
Quantum metrology protocols are typically designed around the assumption that we have an abundance of measurement data, but recent practical applications are increasingly driving interest in cases with very limited data. In this regime the best approach involves an interesting interplay between the amount of data and the prior information. Here we propose a new way of optimising these schemes based on the practically-motivated assumption that we have a sequence of identical and independent measurements. For a given probe state we take our measurement to be the best one for a single shot and we use this sequentially to study the performance of different practical states in a Mach–Zehnder interferometer when we have moderate prior knowledge of the underlying parameter. We find that we recover the quantum Cramér–Rao bound asymptotically, but for low data counts we find a completely different structure. Despite the fact that intra-mode correlations are known to be the key to increasing the asymptotic precision, we find evidence that these could be detrimental in the low data regime and that entanglement between the paths of the interferometer may play a more important role. Finally, we analyse how close realistic measurements can get to the bound and find that measuring quadratures can improve upon counting photons, though both strategies converge asymptotically. These results may prove to be important in the development of quantum enhanced metrology applications where practical considerations mean that we are limited to a small number of trials.
The number of times that we can access a system to extract information via quantum metrology is always finite, and possibly small, and realistic amounts of prior knowledge tend to be moderate. Thus theoretical consistency demands a methodology that departs from asymptotic approximations and restricted parameter locations, while practical convenience requires that it is also flexible and easy to use in applications with limited data. We submit that this methodology can and should be built on a Bayesian framework, and in this thesis we propose, construct, explore and exploit a new non-asymptotic quantum metrology. First we show the consistency of taking those solutions that are optimal in the asymptotic regime of many trials as a guide to calculate a Bayesian measure of uncertainty. This provides an approximate but useful way of studying the non-asymptotic regime whenever an exact optimisation is intractable, and it avoids the non-physical results that can arise when only the asymptotic theory is used. Secondly, we construct a new non-asymptotic Bayesian bound without relying on the previous approximation by first selecting a single-shot optimal quantum strategy, and then simulating a sequence of repetitions of this scheme. These methods are then applied to a single-parameter Mach-Zehnder interferometer, and to multi-parameter qubit and optical sensing networks. Our results provide a detailed characterisation of how the interplay between prior information, correlations and a limited amount of data affects the performance of quantum metrology protocols, which opens the door to a vast set of unexplored possibilities to enhance non-asymptotic schemes. Finally, we provide practical researchers with a numerical toolbox for Bayesian metrology, while theoretical workers will benefit from the broader and more fundamental perspective that arises from the unified character of our methodology.
Science relies on our practical ability to extract information from reality, since processing this information is essential for developing theories that explain our world. This thesis is precisely the study of how to extract and process information using quantum systems when a constrained amount of resources means that the available data is limited. The natural framework for this task is quantum metrology, a set of tools to model and design quantum measurement strategies. Equipped with this theory, we advocate a Bayesian approach as the appropriate formalism to study systems with a finite amount of resources, which is a non-asymptotic problem, and we propose a methodology for non-asymptotic quantum metrology. To start with, we show the consistency of taking those solutions that are optimal in the asymptotic regime of many trials as a guide to calculate a generalised measure of uncertainty in the Bayesian framework. This provides an approximate but useful way of studying the non-asymptotic regime whenever a direct Bayesian optimisation is intractable, and it avoids non-physical results that can arise when only the asymptotic theory is employed. Secondly, we construct a new non-asymptotic Bayesian bound without relying on the previous approximation by first selecting the optimal quantum strategy for a single shot, and then simulating a sequence of repetitions of this scheme, which is suitable for experiments where we do not wish or cannot correlate different trials. These methods are applied to a Mach-Zehnder interferometer, which is a single-parameter problem, and to quantum sensing networks where the nodes are either qubits or optical modes, which are multi-parameter protocols. Our results provide a detailed characterisation of how the interplay between prior information, entanglement and a limited amount of data affects the performance of quantum metrology protocols, which has important implications for the analysis of theory and experiments in this field.
A longstanding problem in quantum metrology is how to extract as much information as possible in realistic scenarios with not only multiple unknown parameters, but also limited measurement data and some degree of prior information. Here we present a practical solution to this: We derive a Bayesian multi-parameter quantum bound, construct the optimal measurement when our bound can be saturated for a single shot, and consider experiments involving a repeated sequence of these measurements. Our method properly accounts for the number of measurements and the degree of prior information, and we illustrate our ideas with a qubit sensing network and a model for phase imaging, clarifying the nonasymptotic role of local and global schemes. Crucially, our technique is a powerful way of implementing quantum protocols in a wide range of practical scenarios that tools such as the Helstrom and Holevo Cram\'er-Rao bounds cannot normally access.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.
Many results in the quantum metrology literature use the Cramér-Rao bound and the Fisher information to compare different quantum estimation strategies. However, there are several assumptions that go into the construction of these tools, and these limitations are sometimes not taken into account. While a strategy that utilises this method can considerably simplify the problem and is valid asymptotically, to have a rigorous and fair comparison we need to adopt a more general approach. In this work we use a methodology based on Bayesian inference to understand what happens when the Cramér-Rao bound is not valid. In particular we quantify the impact of these restrictions on the overall performance of a wide range of schemes including those commonly employed for the estimation of optical phases. We calculate the number of observations and the minimum prior knowledge that are needed such that the Cramér-Rao bound is a valid approximation. Since these requirements are state-dependent, the usual conclusions that can be drawn from the standard methods do not always hold when the analysis is more carefully performed. These results have important implications for the analysis of theory and experiments in quantum metrology.