Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation

Abstract

The quantum multiparameter estimation is very different with the classical multiparameter estimation due to Heisenberg's uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inefficiency of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenberg's uncertainty principle into quantum multiparameter estimation by giving a tradeoff relation between the measurement inefficiencies for estimating different parameters. For pure quantum states, this tradeoff relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the tradeoff between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the tradeoff relation. We also show that our approach can be readily used to derive the tradeoff between the errors of jointly estimating the phase shift and phase diffusion without explicitly parameterizing quantum measurements.

Generalization of Rayleigh's criterion on parameter estimation with incoherent sources

“…with ∂ µ := ∂/∂θ µ being defined for short. Despite that the QCRB is not guaranteed to be attainable in general for the joint estimation of multiple parameters, the QFI matrix still reflects lots of information about the quantum limit of estimation errors [38][39][40][41][42][43][44]. Gauge symmetry.-Before calculating the QFI matrix, observe that, due to the incoherent characteristic of the model, the density operator in Eq.…”