## 3 Results
### Expectation Values  
Subsequently, in order to test the predictions of PM regarding the possibility of extracting the expectation value \(\langle A\rangle\) even from a single detection event, we evaluated \(\langle A\rangle\) with both the PM and PJ methods. Plots in Figure 4a,c,e show the results obtained for the three states in the PJ case. We can see that the photons accumulate around the two eigenvalues' positions \(x_{H}\) and \(x_{V}\), hence, we can statistically find the expectation value as the counts ratio in Equation (8). Plots in Figure 4b,d,f host, instead, the PM results, in which all the photons accumulate in a specific position corresponding to \(x=a\langle A\rangle\), in agreement with our expectations. This is clear evidence that, with PM, each single photon carries information about the expectation value of its polarization. The extracted expectation values are reported in Table 2, column 3 for PJ and in Table 2, column 4 for PM, together with the associated experimental uncertainties (a detailed description of the expectation values and uncertainties analysis can be found in Appendix A). Within the experimental uncertainties, the expectation values extracted with the PM method are in good agreement with the ones obtained with the traditional PJ one, as well as with the theoretical expectations. In [57], a general comparison was performed between the uncertainties associated with the PM and PJ protocols (considering the same amount of initial photons), showing how PM performs better than PJ in terms of precision for almost every initial state. To further confirm this result, the two plots in Figure 5 show the equivalent of the distributions in Figure 4a,b for a few detection events. However, for the PM in Figure 5b, all the counts (except for dark counts) belong to a region centered in the \(x\) axis position corresponding to the expectation value \(\langle A\rangle\), though the same does not happen for the PJ in Figure 5a. Henceforth, with PM, one can achieve a sound estimate of the average polarization value for the state \(\left|\frac{17}{60}\pi\right\rangle\) from just the first detection event (yellow pixel in Figure 5b), obtaining \(\langle A\rangle=-0.3\pm 0.3\), where the uncertainty is estimated from the width of the initial (Gaussian) spatial photon distribution. This result is in agreement with the theoretically expected value \(\langle A\rangle^{\text{th}}=-0.208\). On the contrary, it is not possible to do the same in the PJ case, since the first detection event (yellow pixel in Figure 5a) does not provide any reliable information about the expectation value of the detected photon \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**State** & \(\langle A\rangle^{\text{th}}\) & \(\langle A\rangle^{\text{PI}}\) & \(\langle A\rangle^{\text{PM}}\) \\ \hline \(\left|+\rangle\right.\) & 0 & \(-0.03\pm 0.04\) & \(0.012\pm 0.014\) \\ \(\left|\frac{17}{60}\pi\right\rangle\) & \(-0.208\) & \(-0.21\pm 0.02\) & \(-0.19\pm 0.02\) \\ \(\left|\frac{1}{8}\right\rangle\) & 0.707 & \(0.72\pm 0.02\) & \(0.72\pm 0.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the experimental and theoretical expectation values. Experimental expectation values—\(\langle A\rangle^{\text{th}}\): theoretical expectation values; \(\langle A\rangle^{\text{PI}}\): experimental expectation value with projective measurements; and \(\langle A\rangle^{\text{PM}}\): experimental expectation value with projective measurements. polarization. This is a final demonstration of the PM capability of extracting the expectation value even from a single detection event. Figure 4: Plots of the photon counts distributions obtained with projective (**a,c,d**) and protective (**b,d,f**) measurements, for three different linearly polarized initial states \(|\psi_{\theta}\rangle\).