Measurements of quantum states form a key component in quantum-information processing. It is therefore an important task to compare measurements and furthermore decide if a measurement strategy is optimal. Entropic quantities, such as the quantum Fisher information, capture asymptotic optimality but not optimality with finite resources. We introduce a framework that allows one to conclusively establish if a measurement is optimal in the non-asymptotic regime. Our method relies on the fundamental property of expected errors of estimators, known as risk, and it does not involve optimisation over entropic quantities. The framework applies to finite sample sizes and lack of prior knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go theorem that shows that only classical states admit optimal measurements under the most common choice of error measurement: least squares. We further consider the less restrictive notion of an approximately optimal measurement and give sufficient conditions for such measurements to exist. Finally, we generalise the notion of when an estimator is inadmissible (i.e. strictly worse than an alternative), and provide two sufficient conditions for a measurement to be inadmissible.
Measurements of quantum states form a key component in quantum-information processing. It is therefore an important task to compare measurements and furthermore decide if a measurement strategy is optimal. Entropic quantities, such as the quantum Fisher information, capture asymptotic optimality but not optimality with finite resources. We introduce a framework that allows one to conclusively establish if a measurement is optimal in the non-asymptotic regime. Our method relies on the fundamental property of expected errors of estimators, known as risk, and it does not involve optimisation over entropic quantities. The framework applies to finite sample sizes and lack of prior knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go theorem that shows that only classical states admit optimal measurements under the most common choice of error measurement: least squares. We further consider the less restrictive notion of an approximately optimal measurement and give sufficient conditions for such measurements to exist. Finally, we generalise the notion of when an estimator is inadmissible (i.e. strictly worse than an alternative), and provide two sufficient conditions for a measurement to be inadmissible.
We present system-theoretic quantum state reconstruction methods that minimize estimation error by combining optimal quantum control with asymptotically efficient estimation. Introducing the notion of optimal observability of a quantum dynamical system---a concept that does not exist in classical control theory---we formulate and solve the Pareto optimal control problem of maximizing state estimation accuracy while minimizing the expenditure of available control and measurement resources. Necessary and sufficient conditions for optimal observability, based on the quantum optimal observability Gramian, are presented. We examine the finite sample efficiency of the estimation methodology for two- and three-level systems using ideal and noisy control fields, and demonstrate the advantages of state reconstruction schemes based on optimal observability theory for experimentally realistic sample sizes. These results indicate that the optimal control of quantum measurement bases can be used to minimize state reconstruction errors by fully exploiting the information geometry of quantum states.
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.
We discuss the problem of finding the best measurement strategy for estimating the value of a quantum system parameter. In general the optimum quantum measurement, in the sense that it maximizes the quantum Fisher information and hence allows one to minimize the estimation error, can only be determined if the value of the parameter is already known. A modification of the quantum Van Trees inequality, which gives a lower bound on the error in the estimation of a random parameter, is proposed. The suggested inequality allows us to assert if a particular quantum measurement, together with an appropriate estimator, is optimal. An adaptive strategy to estimate the value of a parameter, based on our modified inequality, is proposed.
We undertake a detailed study of the performance of maximum likelihood (ML) estimators of the density matrix of finite-dimensional quantum systems, in order to interrogate generic properties of frequentist quantum state estimation. Existing literature on frequentist quantum estimation has not rigorously examined the finite sample performance of the estimators and associated methods of hypothesis testing. While ML is usually preferred on the basis of its asymptotic properties - it achieves the Cramer-Rao (CR) lower bound - the finite sample properties are often less than optimal. We compare the asymptotic and finite-sample properties of the ML estimators and test statistics for two different choices of measurement bases: the average case optimal or mutually unbiased bases (MUB) and a representative set of suboptimal bases, for spin-1/2 and spin-1 systems. We show that, in both cases, the standard errors of the ML estimators sometimes do not contain the true value of the parameter, which can render inference based on the asymptotic properties of the ML unreliable for experimentally realistic sample sizes. The results indicate that in order to fully exploit the information geometry of quantum states and achieve smaller reconstruction errors, the use of Bayesian state reconstruction methods - which, unlike frequentist methods, do not rely on asymptotic properties - is desirable, since the estimation error is typically lower due to the incorporation of prior knowledge.
We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.
In quantum estimation for a d-parameter family of density operators on a finite-dimensional Hilbert space H, an estimator is specified by a pair (M,θ^), where M is a POVM with a finite outcome set Ω and θ^:Ω→Rd is a classical estimator map. Since the number of outcomes ∣Ω∣ is a priori unbounded, the space of admissible POVMs is vast, which makes the search for optimal estimators difficult. In this paper, for the minimization of the weighted trace of the mean squared error among locally unbiased estimators, we prove that it suffices to consider POVMs with at most (dimH)2+d(d+1)/2−1 outcomes, and that the optimization can be restricted to rank-one measurements. For Bayesian estimation with a general loss function, we show that rank-one POVMs with at most (dimH)2 outcomes are sufficient for the infimum value of the Bayes risk. Furthermore, when the model admits a real sufficient subalgebra, we show that the (dimH)2 term in the above support-size bounds can be reduced in both the locally unbiased and Bayesian settings. These bounds substantially reduce the search space for optimal measurements and justify restricting numerical optimization to rank-one POVMs with finitely many outcomes.