Quantum Metrology of Noisy Spreading Channels

We provide the optimal measurement strategy for a class of noisy channels that reduce to the identity channel for a specific value of a parameter (spreading channels). We provide an example that is physically relevant: the estimation of the absolute value of the displacement in the presence of phase randomizing noise. Surprisingly, this noise does not affect the effectiveness of the optimal measurement. We show that, for small displacement, a squeezed vacuum probe field is optimal among strategies with same average energy. A squeezer followed by photodetection is the optimal detection strategy that attains the quantum Fisher information, whereas the customarily used homodyne detection becomes useless in the limit of small displacements, due to the same effect that gives Rayleigh’s curse in optical superresolution. There is a quantum advantage: a squeezed or a Fock state with $N$ average photons allow to asymptotically estimate the parameter with a $\sqrt{N}$ better precision than classical states with same energy.

Figure 1

Classical Fisher information for the measurement (3) and the fidelity (inset) between the initial state $|\psi ⟩$ and the final state $|{\psi }_{\alpha }^{\phi }⟩=D(\alpha ,\phi )|\psi ⟩$ (with $D$ displacement), averaged over $\phi$ as a function of $\alpha$. Blue dotted, Fock state; yellow solid, squeezed vacuum state; green dashed, coherent state; all with $N=5$ average photons. For the vacuum state the fidelity is the same as for coherent state (green dashed).

Figure 2

Wigner function of the multi-cat state $|{\mathrm{cat}}_{N,\theta }⟩$ with $N=10$ and 10 average photons, i.e., $\beta \simeq 8.3045$ in Eq. (c1). It is evident that such a state closely approximates a Fock state with the same number of photons (inset).

Figure 3

Plot of the overlap between the initial state and the initial state displaced by $D(\alpha ,\phi )$ (here with fixed phase $\phi$) as a function of $\alpha$. Namely, $\text{overlap}=|⟨\psi |\mathcal{D}(\alpha )|\psi ⟩|$ where $|\psi ⟩$ is any of the states we consider here. The squeezed vacuum state $|r,\theta ⟩$ (green stars with dash-dotted line) performs worse if the squeezing phase $\theta$ is parallel to the displacement phase $\phi$ (upper curve) and it performs very well if the two phases are orthogonal (lower curve). The coherent state, the vacuum state, and the two-cat state with phase $\theta$ parallel to the displacement all perform equally (higher solid multicolored curve). They all perform quite badly: the overlap decreases very little as a function of $\alpha$. The multi-cat state $|{\mathrm{cat}}_{N,\theta }⟩$ (here we use $N=10$ cats) is plotted as the continuous blue curve for parallel phase $\theta$ and as the orange dot-dashed curve for orthogonal phase $\theta$ (they are indistinguishable in this plot as there is little difference between their performance: the multi-cat is almost insensitive to phase). In contrast, the two-cat state is quite sensitive to the phase $\theta$: the cyan dotted curve plots its performance when its phase is orthogonal to the displacement (good behavior), whereas it behaves exactly as the coherent state if the phase is parallel. The best state overall appears to be the two-cat state (but is highly phase sensitive). In contrast, the Fock state (black line) is basically identical to the multi-cat state and are both independent of the phase (the Fock state is exactly independent, the multi-cat state is approximately independent). All states here (except for the vacuum state) have the same average number of photons, $⟨a^{†}a⟩=10$.

Figure 4

Monte Carlo simulation of the overlap between the initial state and the initial state displaced by $D(\alpha ,\phi )$ with randomly chosen $\phi$ as a function of $\alpha$. The coherent and vacuum states (red stars), the Fock state (black lines) and the ten-cat state (blue circles) are unaffected (the multi-cat state is affected slightly, but the effects cannot be seen at this scale). The two-cat state (green squares) is quite phase sensitive and the fluctuations for random phases can be easily seen (vertical bars). The squeezed vacuum state (pink triangles) is also very sensitive to phase. In the regime $\alpha \to 0$, the Fock state, the squeezed vacuum state, the two-cat state, and the ten-cat state all perform in a similar way. The vertical bars are not the statistical error bars: they are an estimation of the fluctuations (the root mean square of the overlap). (The error bars would be the root mean square divided by the square root of the number of simulated phases.) Here, all states (except the vacuum) have $⟨a^{†}a⟩=10$ and an average over $M=50$ uniformly distributed random values of $\phi$ is performed.