## 2 Experimental Implementation
#### Experimental Setup  
Both PM and PJ experimental setups (Figure 2a,b, respectively) can be divided into three parts. In the first part, single photons are generated by a heralded single-photon source [82; 83]. A mode-locked laser with a second harmonic at 398 nm and a 76 MHz repetition rate pumps a \(10\times 10\times 5\) mm LiIO\({}_{3}\) non-linear crystal, producing signal-idler photon pairs by exploiting Type-I spontaneous parametric down-conversion (SPDC). The generated idler photons (920 nm) are filtered by an interference filter (IF) centered at 920 nm and with a FWHM of 10 nm, coupled to a single-mode fiber(SMF) which addresses them to a Silicon single-photon avalanche diode (SPAD), heralding the presence of the correlated signal photons (702 nm). Signal photons are filtered with an IF centered at 702 nm and with a FWHM of 10 nm, fiber-coupled and addressed to the second part of the setup, where the PM takes place. A Hanbury-Brown and Twiss interferometer allowed us to estimate the quality of our single-photon emission by evaluating the \(\alpha\) parameter [84; 85], obtaining a value of \(\alpha=0.13\pm 0.01\) without any background or dark count subtraction, testifying the high quality of our heralded single-photon source. Figure 1: Theoretical framework (ideal pictorial representation). (a) Projective measurement: the two polarization components are completely separated, with the single photons impinging on two regions of the detector corresponding to the polarization operator eigenvalues \(A=\pm 1\). The expectation value \(\langle A\rangle\) is evaluated as the weighted average of the events, following Equation (8). (b) Protective measurements: all the photons fall in the same region, centered in a position proportional to the polarization expectation value (see Equation (9)). Figure 2: Experimental setups for projective measurement (**a**), extracting the expectation value of an observable by measuring an ensemble of identical particles, and protective measurement (**b**), able to reliably estimate such an expectation value with just a single detection event. Ti:Sa ML laser: titanium–sapphire mode-locked laser; SHG: second harmonic generator; SMF: single-mode fiber; SPAD: single-photon avalanche diode; PBS: polarizing beam splitter; HWP: half-wave plate. In the second part, the PM and PJ are performed. The signal photon produced in the previous stage is decoupled and collimated in a Gaussian spatial mode over a 2 m length. Then, it is initialized (pre-selected) in the polarization state \(\ket{\psi}\) by a polarizing beam splitter (PBS) followed by a half-wave plate (HWP). Finally, in the PJ configuration (Figure 2a), the photon goes through \(K=7\) weak interaction units (we chose the number of units \(K=7\) from practical considerations approximating the ideal case of large \(K\)). Each unit consists of a first 2 mm-long birefringent calcite crystal with an extraordinary (\(e\)) optical axis lying in the \(x\)-\(z\) plane, having an angle of \(\pi/4\) with respect to the \(z\) direction, followed by a second 1.1 mm calcite crystal with the optical \(e\)-axis oriented along the \(y\)-axis. The first crystal shifts the horizontally polarized component of the wavefunction along the transverse direction \(x\), while the second one compensates for the phase and time decoherence induced by the first crystal. The combined effect of all the seven units allows for the complete separation of orthogonal polarizations, reproducing the PJ framework. In the PM scenario (Figure 2b), instead, the protection of the quantum state is implemented, exploiting the quantum Zeno effect, realized by inserting a polarizing plate after each weak interaction unit. The polarizing plate realizes the projection \(\Pi_{\psi}=\ket{\psi}\!\!\bra{\psi}\), projecting the state outgoing the weak von Neumann interaction onto the same polarization of the initial state \(\ket{\psi}\), thus canceling the (small) decoherence induced by the birefringent crystals in each weak interaction unit.