Excessive precision compromises accuracy even with unlimited resources due to the trade-off in quantum metrology

This paper provides a unified definition of precision and accuracy from the perspective of distinguishing neighboring quantum states. We find that the conventional quantum Cramér–Rao bound underestimates the effect of statistical noise, because the biases of parameters were inappropriately ignored. Given that probability estimation is unbiased, defining precision based on probability distributions provides a more accurate approach. This leads to a correction of factor 2 to the traditional precision lower bound. The trade-off between precision and accuracy shows that precision can be further improved by sacrificing accuracy, while it should be restricted by the inherent precision limit determined by the number of samples. The inherent precision limit can reach the Heisenberg scaling even without entanglement resources, which, however, comes at the cost of significantly reduced accuracy. We show that increasing sampling may decrease accuracy when one pursues excessive precision, which indicates that the trade-off should be considered even with unlimited resources.