Quantum Fisher information for general spatial deformations of quantum emitters

We present a framework for the detection and estimation of deformations applied to a grid of sources. Our formalism uses the Hamiltonian formulation of the quantum Fisher information matrix (QFIM) as the figure of merit to quantify the amount of information we have on the deformation matrix. Quantum metrology for grid deformations provides an ideal testbed to examine multi-parameter estimations for arbitrarily parameterised channel evolutions with generally non-commuting Hermitian generators. We generalise the local generator of translations for deformation parameters to multi-parameter estimations and use it to explore how well different deformations can be detected and corrected for. This approach holds for any deformation. We explore the application of our theory to the set of affine geometry maps. Both the configuration of the grid and the properties of the sources help to maximise the sensitivity of the QFIM to changes in the deformation parameters. For the non-multiplicative Hamiltonian parameterisations resulting from grid rotations about any chosen axis, oscillatory dependence of the QFI surfaces for a specific interplay between mutual source separation distances and grid configurations