Evaluating the quantum optimal biased bound in a unitary evolution process

Seeking the available precision limit of unknown parameters is a significant task in quantum parameter estimation. One often resorts to the widely utilized quantum Cramér–Rao bound (QCRB) based on unbiased estimators to finish this task. Nevertheless, most actual estimators are usually biased in the limited number of trials. For this reason, we introduce two effective error bounds for biased estimators based on a unitary evolution process in the framework of the quantum optimal biased bound. Furthermore, we show their estimation performance by two specific examples of the unitary evolution process, including the phase encoding and the SU(2) interferometer process. Our findings will provide an useful guidance for finding the precision limit of unknown parameters. •Two effective error bounds are proposed in the framework of a unitary evolution process.•These two error bounds can show better estimation performance compared with the original optimal biased bound.•The original optimal biased bound can be saturated by the classical optimal biased bound based on the optimal measurement.