We demonstrate a method for super-resolving signals encoded as finite mixtures of bosonic modes using collective measurements that exploit permutation symmetry. Specifically, we use multiple copies of the state ρ of the finite mixture to extract an estimate for the purity of ρ via a spectrum measurement, the weak Schur-sampling measurement. Depending on the outcome we then further fine-grain the measurement to optimally extract an estimate of the relative intensity between the two incoherent mixtures. Our protocol furnishes simultaneous estimates for both the relative intensity and the separation of incoherent signals saturating the multi-parameter Cram\'{e}r-Rao bound, and is robust against misalignment errors. We also provide viable experimental avenues for implementing such collective measurements in different set-ups.
@article{arxiv.2110.00986,
title = {Superresolving collective quantum measurements},
author = {J. O. de Almeida and M. Lewenstein and M. Skotiniotis},
journal= {arXiv preprint arXiv:2110.00986},
year = {2025}
}