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 experiment enough times and that we have certain prior knowledge about the unknown parameter 2 [4,5,18,19] (see footnote 2), and this simplifies the optimisation of the error considerably. Furthermore, the Fisher information has a certain fundamental character. In particular, it can be seen as a distinguishability metric [20] that arises in the expansion of the Bures distance between two infinitesimally close states [3,18]. Moreover, its reciprocal gives us the asymptotic limit for the Bayesian mean square error as a function of the number of repetitions under some fairly general assumptions 3 [5] (see footnote 3), and this is also the case for other approaches that are more conservative than the Cramér-Rao bound too [21,22].Nevertheless, the fact that this technique normally requires many repetitions to be useful is an important drawback to study realistic physical systems such as those previously mentioned. This problem has already been acknowledged in the literature (e.g. in [4, 5, 18, 27]), and several solutions have been proposed. A conceptually simple and straightforward approach consists in using a general measure of uncertainty and estimating how many measurements are needed such that the results predicted by the asymptotic theory are valid, which can always be done numerically [5,28]. In addition, we can rely on numerical techniques such as Monte Carlo simulations [29] or machine learning [30] to perform the optimisation directly, or can simply examine the behavi...
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.
A longstanding problem in quantum metrology is how to extract as much information as possible in realistic scenarios, which typically involve multiple parameters, limited measurement data and some degree of prior information. Here we present a practical strategy for achieving just this. First we derive a new Bayesian multi-parameter quantum bound. We then show how to 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. This approach provides a powerful and useful technique for implementing quantum metrology in a wide-range of practical scenarios as well as gaining new insights about the role of local and global strategies.
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