# Statistical Inference with Quantum Measurements: Methodologies for Nitrogen Vacancy Centers in Diamond

## TL;DR

This paper develops statistical inference methods for analyzing photon count data from nitrogen vacancy centers in diamond, improving parameter estimation accuracy and error analysis in quantum measurements.

## Contribution

It introduces a comprehensive statistical framework for NV center measurements, including Bayesian and frequentist estimators, with practical validation on experimental data.

## Key findings

- Bayesian estimators outperform traditional methods in risk.
- Maximum likelihood risk closely matches the Cramer-Rao bound.
- Experimental validation shows improved parameter estimation accuracy.

## Abstract

The analysis of photon count data from the standard nitrogen vacancy (NV) measurement process is treated as a statistical inference problem. This has applications toward gaining better and more rigorous error bars for tasks such as parameter estimation (eg. magnetometry), tomography, and randomized benchmarking. We start by providing a summary of the standard phenomenological model of the NV optical process in terms of Lindblad jump operators. This model is used to derive random variables describing emitted photons during measurement, to which finite visibility, dark counts, and imperfect state preparation are added. NV spin-state measurement is then stated as an abstract statistical inference problem consisting of an underlying biased coin obstructed by three Poisson rates. Relevant frequentist and Bayesian estimators are provided, discussed, and quantitatively compared. We show numerically that the risk of the maximum likelihood estimator is well approximated by the Cramer-Rao bound, for which we provide a simple formula. Of the estimators, we in particular promote the Bayes estimator, owing to its slightly better risk performance, and straight-forward error propagation into more complex experiments. This is illustrated on experimental data, where Quantum Hamiltonian Learning is performed and cross-validated in a fully Bayesian setting, and compared to a more traditional weighted least squares fit.

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Source: https://tomesphere.com/paper/1705.10897