Experimental function estimation from quantum phase measurements

Characterizing and analyzing a system often requires learning an unknown function, such as the response of a system or the profile of a field. The standard approach is to opportunely sample the function at fiducial points and then interpolate. When the quantity of interest is embodied in physical objects accessible with quantum-enhanced measurements, it becomes relevant to investigate how to transfer this advantage from the individual sampled points to the estimation of the whole function. In this article we report the experimental quantum-enhanced function estimation of the optical response of a liquid crystal. Our results illustrate that optimizing the employment of the resources is not as straightforward as it may appear at a first glance: Quantum advantage becomes substantial only past a sampling density that depends on the interpolation method, and on the function at hand. Our results show how quantum resources should successfully be employed to access the rich information contained in continuous signals.

Figure 1

Conceptual scheme of function estimation. (a) The optical response of a system to an applied voltage $x$ is captured as the phase $\phi \left(x\right)$ introduced between the polarization components aligned along the two optical axes. This phase is estimated at different values of $x$ with suitably prepared light probes. (b) The experiment yields a function $\tilde{\phi }\left(x\right)$ being an estimator of the actual response $\phi \left(x\right)$, based on a set of fiducial points to which a statistical error is associated due to the limited amount of resources per point adopted.

Figure 2

Experimental setup. A CW diode laser at 405 nm pumps a 3-mm $\beta$-barium borate (BBO) crystal cut for noncollinear type-I phase matching, generating via SPDC two degenerate photons at 810 nm. The laser power is set to 50 mW for the reference measurement and to 6 and 13 mW, respectively, for the measurements at $N_r=800$ an $N_r=3800$. The two modes are then selected through interference filters with FWHM = 7.3 nm and single-mode fibers. For the NOON measurements, the polarization of one mode is rotated by means of a half-wave plate (HWP), and the two photons are then overlapped on the same spatial mode using a PBS, generating NOON states in the diagonal polarization basis. The two photons are then sent through the liquid-crystal device which alters the relative phase between the two modes according to the applied voltage. A projective measurement is performed by means of a a second HWP and PBS. For the individual photon measurements, the $|H\rangle$ polarized photon is still sent through the same setup, whereas the other is directly coupled to the detector for heralding. The output modes of the interferometer are coupled to single-mode fibers and sent to two avalanche photodiodes (APDs) for detection. The acquisition time is set to 3 s for the reference measurement and to 0.5 s for the two measurements at lower $N_r$. Coincidences are recorded by means of a field programmable gate array (FPGA).

Figure 3

Response function measurement. (a) Birefringent reference phase ${\phi }_{\mathrm{ref}}\left(x\right)$ imparted by the liquid-crystal device at different voltage values measured with NOON states with $N_s=500(\mathrm{\Delta }x=0.006\mathrm{V})$ and $N_r\simeq 60\times {10}^3$, estimated via a multiparameter Bayesian approach as discussed in the text. In the graph the errors are smaller than the datapoints. (b) Example of the interpolated estimated response function $\tilde{\phi }\left(x\right)$ for $N_s=5$ and $N_r=3800$. The shaded area corresponds to the difference $\mathrm{\Delta }\tilde{\phi }\left(x\right)={\phi }_{\mathrm{ref}}\left(x\right)-\tilde{\phi }\left(x\right)$ for classical resources (red) and NOON states (blue). The complete overlap of the two curves, resulting in the brown colored area, shows that there is no difference at this sampling density in the estimation with quantum and classical resources.

Figure 4

Interpolation error. The three panels show a portion of the interpolated estimated response function $\tilde{\phi }$ obtained with classical (red triangles) and quantum (blue circles) resources with $N_r=3800$ for three different sampling densities: (a) $N_s=5$, (b) $N_s=10$, and (c) $N_s=100$. The shaded area corresponds to the difference $\mathrm{\Delta }\tilde{\phi }\left(x\right)={\phi }_{\mathrm{ref}}\left(x\right)-\tilde{\phi }\left(x\right)$ for classical resources (red) and NOON states (blue). The brown shade indicates where the two contributions overlap.

Figure 5

Function estimation. Simulated (dotted lines) and measured error ${\delta }^2$ as a function of the number of sampled points $N_s$ with NOON states (blue circles) and individual photons (red squares) probes for (a) $N_r=800$ for nearest-neighbor interpolation, (b) $N_r=3800$ for nearest-neighbor interpolation, (c) $N_r=800$ for linear interpolation, and (d) $N_r=3800$ for linear interpolation. The errors are compatible with what expected from the simulations performed at the Cramér-Rao bound. The insets show, on the same scales, the two contributions to ${\delta }^2$: The one arising from the statistical error on the fiducial points, which increases with the number of sampling points, and the contribution due to the interpolation on the remaining points, which instead decreases with the sampling density.

Figure 6

Simulated error ${\delta }^2$ for the LC response function (a). The error ${\delta }^2$ is evaluated with linear (left column) and nearest-neighbor (right column) interpolations with NOON states (light blue, “-.”) and individual photons (red, “- -”) probes for (b) and (c) $\mathbf{N}={10}^3$, (d) and (e) $\mathbf{N}={10}^4$, (f) and (g) $\mathbf{N}=2\times {10}^4$, and (h) and (i) $\mathbf{N}=5\times {10}^4$.

Figure 7

(a) Slow exponential decay, (d) low-frequency sine response function, and (g) high-frequency sine response function. (b), (e), and (h) Simulated error ${\delta }^2$ as a function of the number of sampled points $N_s$ with NOON states (light blue, “-.”) and individual photons (red, “- -”) probes for linear interpolation, and (c), (f), and (i) nearest-neighbor interpolation.