Moment-based superresolution: Formalism and applications

Optimal Observables and Estimators for Practical Superresolution Imaging

“…To calculate the quantities ( 3) and ( 4), we extend the modes u AE ðrÞ to a complete orthonormal basis, and we expand the field operators âk in this basis. By means of this expansion, the mean photon number N k and the correlations h Nk Nl i can be fully expressed in terms of the expectation values h b † [39]. Finally, we obtain ðΓ½d; θ;…”

“…where vAE are field operators associated with auxiliary modes, that we can assume to be in the vacuum state, and κ AE ¼ κð1 AE δÞ. Equation (2) allows us to propagate the quantum state ρ0 of the sources into its image ρðd; θÞ after transmission through the imaging system [39].…”

“…in the image plane, with the information on the parameter d encoded both in the shape of the modes u AE ðrÞ and the mean photon numbers N AE ¼ Nκ AE [39].…”

“…Moreover, let us assume ideal intensity measurements in the Hermite Gaussian (HG) basis v k ðrÞ ¼ u k¼ðn;mÞ ðrÞ with u 00 ðrÞ ¼ u 0 ðrÞ. In this scenario, the covariance matrix (5) can be inverted analytically and it is possible to derive exact expressions for the measurement sensitivity M½d; θ; N and coefficients m for an arbitrary number K of measured modes [39]. In particular, if we measure intensity in the full HG basis, our estimation strategy saturates the quantum Cramér-Rao bound calculated in Refs.…”

“…In particular, if we measure intensity in the full HG basis, our estimation strategy saturates the quantum Cramér-Rao bound calculated in Refs. [19,20], i.e., lim K→∞ M½d; θ; N ¼ F Q ½d; θ [39]. When the number of received photons is low (Nκ ≪ 1), the quantum Fisher information is constant F Q ½d; θ ¼ 2Nκ=w 2 [18], accordingly small distances can be resolved as well as large ones.…”