P T-Enhanced Bayesian Parameter Estimation

The frequentist (local) quantum parameter estimation based on the Cramer-Rao bound is an asymptotic theory and therefore requires an infinite number of samples for the measurements. On the other hand, by introducing the prior probability distribution, Bayesian parameter estimation approach relaxes this requirement and can be successfully applied under realistic assumption of possessing a limited prior information about the parameter value under investigation. Meanwhile, recent advances in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, both theoretically and experimentally, provide an opportunity to significantly enhance the parameter estimation precision by manipulating the Hilbert space of the system. Taking advantage of these, we develop the Bayesian parameter estimation approach for the two-level system possessing ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry. We consider both options available in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, namely the ${{{\mathcal{C}}}}{{{\mathcal{P}}}}{{{\mathfrak{T}}}}$ and Hermitian measurements, for encoding the probe state and manipulation of the quantum Fisher information of the system and take into account possible decoherence effects inherent for the practical implementations. Our approach should be particularly relevant for applications where the accuracy of quantum parameter estimation is crucial, such as quantum randomness sources and the quantum key distribution. Finally, to bridge the gap with the practical implementations, we consider the relation of our scheme with the current implementation of ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry on the IBM quantum processor and formulate our approach as Bayesian parameter estimation employing an ancilla qubit.