Quantum Fisher information for general spatial deformations of quantum emitters. (arXiv:1802.01601v1 [quant-ph])

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 (\textsc{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 \textsc{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 \textsc{qfi}
surfaces for a specific interplay between mutual source separation distances
and grid configurations.

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