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

Abstract: 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.