Sensitivity Bounds for Multiparameter Quantum Metrology

We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes, or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies that saturate the precision bounds are provided.

Figure 1

General scheme for multiparameter quantum metrology with commuting generators of phase shifts. (a) The probe state $\widehat{\rho }$ of $N$ particles is distributed among $M$ modes. In each mode $k=1,\dots ,M$, a parameter ${\theta }_k$ is encoded as a relative phase shift between sublevels. The sensitivity is quantified by the covariance matrix of the estimators $\mathbf{\Sigma }$. The probe state $\widehat{\rho }$ can be prepared as schematically shown in (b): mode and particle separable (MsPs), mode separable and particle entangled (MsPe), mode entangled and particle separable (MePs), and mode and particle entangled (MePe). The gray bars represent the particle partition of the quantum state, the white bars the mode partition. Mode entanglement is illustrated by vertical blue delocalized distributions, particle entanglement by horizontal delocalization.