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 utilizes 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.
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 utilizes 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.
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.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.
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.
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.
In quantum metrology, a major application of quantum technologies, the ultimate precision of estimating an unknown parameter is often stated in terms of the Cramér-Rao bound. Yet, the latter fails to completely characterize the distribution of estimates in the nonasymptotic regime. Optimizing metrology protocols with respect to the Cramér-Rao bound can therefore lead to surprisingly poor finite-sample performance. This can be avoided by quantifying the quality of a metrology protocol by the probability of obtaining an estimate with a given accuracy. Using this intrinsically single-shot quantity naturally accommodates the finite-sample regime. We show that the fundamental limits of this figure of merit can be quantified through a multihypothesis testing problem between quantum states. Building on this connection, we derive an analogue of the Cramér-Rao bound that contains explicit corrections relevant to the finite-sample regime. We apply the finite-sample metrology framework to the example task of phase estimation with an ensemble of spin-1/2 particles, giving unambiguous evidence that large quantum Fisher information does not guarantee that a metrology protocol has a good finite-sample performance. Overall, analyzing the probability of success allows the reliable study of quantum metrology in the finite-sample regime and opens up a plethora of new avenues for research at the interface of quantum information theory and quantum metrology.
Abstract Chapter 24 provides a compact introduction to the topic of quantum metrology, focusing on Hamiltonian parameter estimation in the frequentist and in the Bayesian paradigms. We first discuss how estimates of non-directly measurable quantities such as phases are obtained from measurement statistics in the frequentist approach. We then discuss the Cramér-Rao bound and the Fisher information and study single-qubit phase estimation in the light of this result. We then turn to the multi-qubit setting and discuss the quantum Cramér-Rao bound, for which the quantum Fisher information is the central quantity of interest. In this context we discuss the Uhlmann fidelity in detail. We further contrast Heisenberg scaling with the standard quantum limit in the phase-estimation scenario. Finally, we analyse phase estimation in the Bayesian-estimation approach and derive the van Trees inequality as a Bayesian version of the Cramér-Rao bound