Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry

Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cramér-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.

Figure 1

Schematic of an imaging system: A two-mode $n$-photon quantum probe state $|{\psi }_{\mathrm{in}}\rangle$ irradiates the sample and provides the reference arm of a two-port interferometer. The detection is chosen so as to estimate simultaneously the phase shift $\varphi$ (Hamiltonian dynamics) and the transmission $\eta$ (dissipative dynamics) from the state $\rho (\varphi ,\eta )$ resulting from the interaction of the probe with the sample.

Figure 2

Coefficients of the optimal probe states of the form in Eq. (4) for $n=6$ obtained by numerical optimization. The state resembles the optimal probes for phase estimation in the presence of losses [14] and interpolates towards the Fock state $|6,0\rangle$, optimal state for estimating $\eta$. Solid lines denote simultaneous estimation of phase and loss and dashed lines estimation of only the phase in the presence of losses [14].

Figure 3

Precisions attainable in the combined estimation of loss and phase with different quantum probe states for $n=6$: $n00n$ states (black), Holland-Burnett states (orange), optimal lossy phase estimation states (red), and our optimal states for simultaneous phase and loss estimation (blue). Here $\eta$ is the loss and $\Delta =\sqrt{{\left({\mathcal{I}}_{\varphi \varphi }\right)}^{-1}+{\left(I_{\eta \eta }\right)}^{-1}}$.

Figure 4

Contributions to the combined precision $\Delta \mathit{\theta }$ (blue) separated into the phase $\Delta \varphi$ (red) and loss $\Delta \eta$ (green) parts. The solid and dashed lines represent the precisions for the optimal state for simultaneous phase and loss and for phase estimation, given by the solid lines and dashed lines, respectively, in Fig. 2. All plots are for $n=6$. For comparison, we also report the precisions attainable with a coherent state probe $|\alpha \rangle$ with amplitude ${\left|\alpha \right|}^2=6$ as the dash-dotted line in terms of the standard interferometric limit [14]. All the curves are independent of $\varphi$ and $\Delta \varphi ={\left({\mathcal{I}}_{\varphi \varphi }\right)}^{-1/2}$, $\Delta \eta ={\left(I_{\eta \eta }\right)}^{-1/2}$, and ${\left(\Delta \theta \right)}^2={\left(\Delta \varphi \right)}^2+{\left(\Delta \eta \right)}^2$.

Figure 5

Phase uncertainty $\Delta \varphi$ for the optimal phase estimation scheme (blue) and for the multiparameter scheme (magenta) for photon number $n=5,10,50,100,500$ and $\eta =0.9$.

Figure 6

Probe states with fixed number of photons $n$ for the joint estimation of phase and loss simultaneously (left) and phase only (right). The bars represent the values of $x_k={\left|{\alpha }_k\right|}^2$, with $k=n$ at the top to $k=0$ at the bottom. The topmost bar thus shows the contribution of all the $n$ photons in the lossy arm.