Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation

TL;DR

This paper integrates Heisenberg's uncertainty principle into quantum multiparameter estimation, establishing a tradeoff relation for measurement inaccuracies that reveals fundamental quantum limits and guides optimal joint measurements.

Contribution

It introduces a novel tradeoff relation linking measurement inaccuracies for different parameters, incorporating Heisenberg's principle into quantum estimation theory.

Findings

The tradeoff relation is tight for pure states, revealing true quantum limits.

Derived the tradeoff between real and imaginary parts of a complex signal in coherent states.

Showed how to obtain joint measurement errors for phase shift and phase diffusion without explicit parameterization.

Abstract

The quantum multiparameter estimation is very different from 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 inaccuracy 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 inaccuracies 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.