Experimental ancilla-assisted phase estimation in a noisy channel

The metrological ability of carefully designed probes can be wrecked by the presence of noise which alters their evolution, reducing the information on the parameter of interest. This spoils the signal-to-noise ratio of any possible measurement. Here we show an experiment in which the introduction of an ancilla improves the estimation of an optical phase with respect to a standard single-qubit strategy. The advantage originates from the coherent coupling of the probe and the ancilla both at the initialization and measurement stages, generating and selecting a subset of overall configurations less affected by the noise.

Figure 1

Illustration of the protocol. (a) The standard estimation procedure entails probe preparation, its interaction with the system to be sampled, and its measurement. The estimation is spoilt by the presence of noise, which reduces the available information in the probe quantum state; hence a large uncertainty will be associated to the parameter $\varphi$. (b) Ancilla-assisted estimation considering a joint probe-ancilla preparation. The probe interacts only with the sample and is hence subject to noise, while the ancilla is left unchanged. Finally, a joint probe-ancilla measurement is performed; this delivers a smaller uncertainty on $\varphi$ with respect to the previous case.

Figure 2

Experimental setup. A 405-nm CW diode laser undergoes SPDC on a 3-mm BBO type-I crystal. One of the two photons produced is used as a trigger, while the other feeds a Sagnac interferometer in which one of the mirrors is replaced by a SLM. The polarization of the input state is set on $|D\rangle$. The SLM and two of the HWPs at $22.5^{\circ }$, constituting the amplitude damping channel, are used to implement the polarization-path coupling. A tunable phase mask is applied on the SLM, allowing the amount of noise in the channel to be varied. A third HWP provides a unitary operation $U_{\varphi }$, used to set the phase to be estimated. The polarization of the output states of the Sagnac are analyzed by means a quarter-wave plate (QWP), HWP, and a PBS. Coincidences between the outputs and the trigger are detected by means of avalanche photodiodes (APD).

Figure 3

Statistical distribution for the outcomes of the four outputs of the setup (see main text, Fig. 2) for different levels of noise ($\eta$). The red histogram represents the measured count occurrences, and the black dashed line is the expected Poissonian distribution.

Figure 4

Results. We report the behavior of the standard deviation (SD) of the phase as a function of the channel noise $\eta$. The red dotted curve represents the theoretical SD (best performance) for the case of a single-qubit estimation measurement. The green dashed line represents the theoretical SD for the ancilla-assisted case, accounting for the reduced visibility of the interferometer ($v=0.9860\pm 0.0003$). Data are reported as follows: orange open circles with error bars represent the phase estimation based on the repetition of the experiment, black open diamonds come from the Bayesian estimation (error bars are slightly smaller than the point size), and the Monte Carlo simulation is reported as a red cross. Horizontal error bars are significantly smaller than the point dimension. Experimental data confirm an effective advantage of the ancilla-assisted strategy in the high-noise regime.

Figure 5

Density probability distribution for the parameter $\varphi$ for two distinct noise levels $\eta$. The experimental ancilla-assisted Bayesian a posteriori distribution (solid line) is compared to the ideal case for single-qubit measurement (dashed line). The narrowing of the distribution in the high-noise regime is evident.