Figure 1

Schematic diagram of the measurement scheme.Reuse & Permissions

Figure 2

Effect of the phase-destroying ME on the Bloch sphere: the $z$ component is left unchanged while the $x$ and $y$ components are scaled by the factor $e^{-{\Delta }^2}$. See the text for details.Reuse & Permissions

Figure 3

The distance $\overline{\mathit{ab}}$ on the Bloch sphere between two equatorial states separated by a phase shift, $\varphi ,$ is larger than the distance $\overline{a^{\prime }{b}^{\prime }}$ between nonequatorial states separated by the same amount, $\varphi$. The same conclusion holds in the presence of phase noise: best estimation is achieved by involving equatorial states.Reuse & Permissions

Figure 4

Plot of the Fisher information $F(\varphi ,{\Delta }^2)$ in the case of equatorial qubit probe states ($\theta =\pi /4$) as a function of $\delta =\alpha -\varphi$ and different values of ${\Delta }^2$.Reuse & Permissions

Figure 5

Sensitivity of ${\sigma }_x$ measurement to a small change $d\varphi$ in the phase shift. The projections onto the $x$ axis are the expectations $\langle {\Theta }_0\rangle$, with ${\Theta }_0={\sigma }_x$. For a fixed change $d\varphi$, the change of $\langle {\Theta }_0\rangle$ at $\varphi =0$ is smaller than the one at $\varphi =\pi /2$. For the sake of simplicity we sketch only the upper right quarter of the Bloch sphere $x$-$y$ equatorial plane.Reuse & Permissions

Figure 6

Variances from simulated Monte Carlo experiments performed with $M=60$ copies of the input state and $\Delta =0.34$. The plots on the top refer to the inversion method, the bottom ones to Bayesian analysis. For the plots on the left we chose a coherent state with an average number of photons equal to $\overline{n}=2$, for those on the right we set $\overline{n}=12$. The solid line is the Cramér-Rao bound $1/(FM\overline{n})$. The dashed horizontal line is the quantum Cramér-Rao bound $1/(HM\overline{n})$. Notice the asymptotic optimality of the Bayesian estimation.Reuse & Permissions

Figure 7

Sketch of the experimental setup to estimate the phase of a polarization qubit in the presence of nondissipative noise. The qubit state is set to $|+\rangle$ by inserting a half-wave plate (HWP1) along the path of the He:Ne laser output, while a KDP crystal, driven by a high-voltage generator (HV), is used both to add the phase shift $\varphi$ and to simulate the noise. A second half-wave plate at $22.5^{°}$ (HWP2) together with a polarizing beam splitter (PBS) implements the measurement (the couplers, C, and the multimode fibers, MF, are used to bring the signals to the avalanche photodiodes, APD). Finally, the contemporary counts $N_{1g}$ and $N_{2g}$ coming from the APD1 and APD2 and from the APDg, that is, the gate channel signal coming from the beam splitter (BS) placed in front the laser source, are acquired by a PC module.Reuse & Permissions

Figure 8

Variances from experiments performed with $M=60$ copies of the input state and (a) $\Delta =0.13\pm 0.02$, (b) $\Delta =0.24\pm 0.03$, and (c) $\Delta =0.48\pm 0.05$. Red squares refer to inversion method; blue circles refer to Bayesian analysis. We also have (a) $\overline{n}=9.83\pm 0.93$, (b) $\overline{n}=11.06\pm 0.43$, and (c) $\overline{n}=9.78\pm 0.95$. The solid line is the Cramér-Rao bound $1/(FM\overline{n})$.Reuse & Permissions