Superresolved Localisation of Blinking Sources of Light and Convexity of Fisher Information Matrix

In the last two decades, a number of techniques have been developed for localisation of stochastically activated (blinking) fluorophores such as photoactivated localisation microscopy (PALM), stochastic optical reconstruction microscopy (STORM), points accumulation for imaging in nanoscale topography (PAINT), and their variants. These techniques are know by the general name of single-molecule localisation microscopy (SMLM) and have found numerous applications in cell biology [1]. These techniques can be considered in the framework of the quantum theory of multiple parameter estimation [2]. In this theory, the ultimate limit of precision is set by the quantum Cramér-Rao bound, Cov <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\hat{\theta})\geq \mathbf{F}^{-1}\geq \mathbf{Q}^{-1}$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\hat{\theta}$</tex> is an unbiased estimator for the vector of parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\theta=(\theta_{1}, \ldots, \theta_{M})$</tex>, Cov stands for covariance matrix, and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M\times M$</tex> matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{F}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{Q}$</tex> are the Fisher information matrix and quantum Fisher information matrix, respectively.