Continuous-variable phase estimation with unitary and random linear disturbance

We address the problem of continuous-variable quantum phase estimation in the presence of linear disturbance at the Hamiltonian level by means of Gaussian probe states. In particular we discuss both unitary and random disturbance by considering the parameter which characterizes the unwanted linear term present in the Hamiltonian as fixed (unitary disturbance) or random with a given probability distribution (random disturbance). We derive the optimal input Gaussian states at fixed energy, maximizing the quantum Fisher information over the squeezing angle and the squeezing energy fraction, and we discuss the scaling of the quantum Fisher information in terms of the output number of photons, $n_{\mathsf{out}}$. We observe that, in the case of unitary disturbance, the optimal state is a squeezed vacuum state and the quadratic scaling is conserved. As regards the random disturbance, we observe that the optimal squeezing fraction may not be equal to one and, for any nonzero value of the noise parameter, the quantum Fisher information scales linearly with the average number of photons. Finally, we discuss the performance of homodyne measurement by comparing the achievable precision with the ultimate limit imposed by the quantum Cramér–Rao bound.

Figure 1

QFI, $H$, as a function of the output number of photons $n_{\mathsf{out}}$. Solid line is the noiseless estimate $\left(\eta =0\right)$; dashed line is the estimate with unitary disturbance for different values of $\eta$. From top to bottom, $\eta =\{0.5,1,1.5\}$.

Figure 2

Ratio between the homodyne detection FI, $F$, and the corresponding QFI, $H$. (a) $F/H$ as function of the number of photons of the probe for different values of disturbance $\eta$; using the right ends of the curves as reference, from top to bottom: $\eta =\{0.0,0.25,0.50,0.75,1.0\}$. (b) $F/H$ as function of the disturbance parameter for different values of the average number of photons of the probe; from top to bottom: $n_0=\{0,0.2,0.4,0.6,0.8,1.0\}$.

Figure 3

Optimal squeezing fraction ${\beta }_{\mathsf{opt}}$ as function of the noise parameter $\Delta$ and number of photons $n_0$ in the probe. The threshold ${\Delta }_c\left(n_0\right)$ is represented with a black curve.

Figure 4

(a) Optimized QFI in the case of random linear disturbance as function of the noise parameter $\Delta$ for fixed values of the average photon number; from bottom left to top right $n_0=\{1,21,41,61,81,101\}$. (b) Optimized QFI as a function of the output photon number $n_{\mathsf{out}}$ for fixed values of the noise parameter $\Delta$; from top to bottom $\Delta =\{0.7,0.75,0.8,0.85,0.9,0.95,1.0\}$.

Figure 5

Ratio between the Fisher Information of the homodyne detection and the corresponding QFI for squeezed vacuum input states that have undergone a random linear disturbance. The ratio is plotted as a function of the noise parameter $\Delta$ and for different values of the average number of photons of the probe. From top to bottom: $n_0=\{{10}^{-9},2,4,6,8,10\}$.

Figure 6

Ratio $F/H$ between the homodyne's FI over the QFI for optimized input state as a function of the input photon number $n_0$ and of the noise parameter $\Delta$. The superimposed black line corresponds to the input state threshold ${\Delta }_t\left(n_0\right)$ which divides regions with squeezed vacuum probes $\mathrm{[}\Delta <{\Delta }_t\left(n_0\right)\text{]}$ and displaced squeezed probes $\mathrm{[}\Delta >{\Delta }_t\left(n_0\right)\text{]}$.