Cost-effective temperature estimation strategies for thermal states with probabilistic quantum metrology

In probabilistic quantum metrology, one aims at finding weak measurements that concentrate the Fisher Information on the resulting quantum states, post-selected according to the weak outcomes. Though the Quantum Cram\'er-Rao bound itself cannot be overshot this way, it could be possible to improve the information-cost ratio, or even the total Fisher Information. We propose a post-selection protocol achieving this goal based on single-photon subtraction onto a thermal state of radiation yielding a greater information-cost ratio for the temperature parameter with respect to the standard strategy required to achieve the Quantum Cram\'er-Rao bound. We address just fully-classical states of radiation: this contrasts with (but does not contradict) a recent result proving that, concerning unitary quantum estimation problems, post-selection strategies can outperform direct measurement protocols only if a particular quasiprobability associated with the family of parameter-dependent quantum states becomes negative, a clear signature of nonclassicality.