Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases

A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this Letter, we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the ultimate theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.

Figure 1

General framework of multiparameter estimation considered in this Letter: A probe state $|\psi ⟩$ is prepared, transformed according to the unitary parameter-dependent transformation $\widehat{U}(\mathit{\theta })$, and measured by a set of projectors $\{{\widehat{\mathrm{\Pi }}}_k\}$. The vector parameter is retrieved via an estimator $\mathrm{\Theta }(\mathit{k})$, which is a function of the measurement outcomes.

Figure 2

$\parallel \mathit{F}-{\mathit{F}}_{\mathit{Q}}{\parallel }_2$ of a three-mode (a) and a four-mode (b) interferometer (see the text and Ref. [45]) for the estimation of $\mathit{\theta }=({\theta }_1,{\theta }_2)$. For the three-mode interferometer, there is no value of $\mathit{\theta }$ where the theorems are satisfied. Consistently, we find $\parallel \mathit{F}-{\mathit{F}}_{\mathit{Q}}{\parallel }_2>3/4$. For the four-mode interferometer, we can find values of $\mathit{\theta }$ where the saturation conditions $\mathit{F}(\mathit{\theta })={\mathit{F}}_{\mathit{Q}}(\mathit{\theta })$ discussed in the Letter are fulfilled: These are given by the dashed line and circles.