## 3 Results
### Output State Verification  
To immediately highlight the difference between PM and PJ, we performed a tomographic reconstruction of the states at the end of the measurement process.  
In the PJ case, we expected that the repeated shifts of the horizontal polarization component would cause decoherence on \(\hat{\rho}_{in}=\ket{\psi}\!\!\bra{\psi}\), generating a final state \(\hat{\rho}_{dec}\). Incontrast, in the PM case, the protection should be able to preserve, in principle, the initial polarization state \(\hat{\rho}_{in}\), thus we expect a final state \(\hat{\rho}_{prot}=\hat{\rho}_{in}\):  
\[\hat{\rho}_{in} =\begin{pmatrix}\cos^{2}(\theta)&-\sin(\theta)\cos(\theta)\\ \sin(\theta)\cos^{(}\theta)&\sin^{2}(\theta)\end{pmatrix} \tag{10}\] \[\hat{\rho}_{dec} =\begin{pmatrix}\cos^{2}(\theta)&-\sin(\theta)\cos(\theta)\exp \biggl{(}-\frac{\sigma^{2}}{(2\sigma)^{2}}\biggr{)}\\ \sin(\theta)\cos(\theta)\exp\biggl{(}-\frac{\sigma^{2}}{(2\sigma)^{2}}\biggr{)} &\sin^{2}(\theta)\end{pmatrix} \tag{11}\]  
where \(g^{\prime}=\langle x_{H}\rangle-\langle x_{V}\rangle=11.56\pm 0.07\) px (pixels), as \(\langle x_{H(V)}\rangle\) is the average position of photons in the \(H(V)\) polarization on the \(x\) axis, and \(\sigma=4.17\pm 0.02\) px is the distribution width obtained by Gaussian fits.  
We first computed the distance between the reconstructed states \(\rho_{PM}^{rec}\) and \(\rho_{PI}^{rec}\), respectively, obtained in the PM and PJ case (shown in Figure 3 for the state \(\ket{\frac{17}{60}\pi}\)), and their theoretical counterparts \(\rho_{in}\) and \(\rho_{dec}\), by evaluating the Fidelity \(F\)[89; 90] between them (second and third columns of the Table 1). The high fidelities obtained certify the adherence of the experimental results to our model, showing that PM indeed preserves the initial state by the decoherence induced by the birefringent crystals, while this does not happen for PJ.  
Then, we compute the distance between the two reconstructed states (fourth column of Table 1). Again, the low fidelities obtained tell us that, without protection, the decoherence induced on the initial state by the \(K=7\) unitary interactions makes the final state totally incompatible with the one outgoing the PM procedure (and, obviously, with the initial state itself).  
Finally, by analyzing the purity \(\mathcal{P}\)[89] of the reconstructed states (last two columns of Table 1), we notice that, as expected, the decoherence reduced the purity of the initial state \(\rho_{in}\) in the PJ case, while this does not happen in the PM one. Thus, we proved the PM protocol's ability to preserve the coherence of the initial state during the whole measurement process, a feature in sharp contrast with the traditional quantum measurement paradigms.  
Figure 3: Density matrix reconstructions for the state outgoing the measurement process, considering an initial polarization state \(\ket{\frac{17}{60}\pi}\): (**a**) theoretical real part \(\mathrm{Re}[\rho_{in}]\) of the initial state density matrix (\(\mathrm{Im}[\rho_{in}]=0\)); (**b**,**c**), respectively: the reconstructed real (\(\mathrm{Re}[\rho_{PM}^{rec}]\)) and imaginary (\(\mathrm{Im}[\rho_{PM}^{rec}]\)) parts of the density matrix of the single-photon state after the protective measurement; (**d**) theoretically expected real part \(\mathrm{Re}[\rho_{dec}]\) of the density matrix of our state at the end of the projective measurement (\(\mathrm{Im}[\rho_{dec}]=0\)); and (**e**,**f**), respectively: reconstructed real (\(\mathrm{Re}[\rho_{PI}^{rec}]\)) and imaginary (\(\mathrm{Im}[\rho_{PI}^{rec}]\)) parts of the density matrix of the single-photon state after undergoing the projective measurement.