Quantum limits on postselected, probabilistic quantum metrology

Probabilistic metrology attempts to improve parameter estimation by occasionally reporting an excellent estimate and the rest of the time either guessing or doing nothing at all. Here we show that probabilistic metrology can never improve quantum limits on estimation of a single parameter, both on average and asymptotically in number of trials, if performance is judged relative to mean-square estimation error. We extend the result by showing that for a finite number of trials, the probability of obtaining better estimates using probabilistic metrology, as measured by mean-square error, decreases exponentially with the number of trials. To be tight, the performance bounds we derive require that likelihood functions be approximately normal, which in turn depends on how rapidly specific distributions converge to a normal distribution with number of trials.

Figure 1

Summary of the relationships among the states in our analysis: the diagram should be read as a threat advisory, where the danger is decreased Fisher information. Our main results are that the Fisher information cannot increase as one goes from the top of the diagram to the bottom; the decrease in going to the bottom state is statistical in a sense analyzed further in Sec. 5. The boxes in the right column give the inequalities between the Fisher informations and cite which of the principles enunciated in Sec. 4 encapsulates why the Fisher information decreases.