## 4 Conclusions  
Protective measurements represent a novel, groundbreaking measurement paradigm, which allow preserving the initial coherence of the measured state and extracting the expectation value of an observable, to date considered as a purely statistical quantity, even from a single detection event. The presented results demonstrate this unprecedented capability by exploiting certified single photons. In particular, in this paper, we described in detail the first experimental implementation of PM, providing the readers with all the details needed for a full understanding of the experiment and obtained results, as well as of the related implications for quantum mechanics foundations. We verified that PM preserves the coherence of the initial state during the whole measurement process, as certified by the high fidelities between the initially prepared states and the reconstructed ones outgoing the PM. Furthermore, the ability of PM to extract the expectation value of a quantum observable from a single (protected) particle is demonstrated by the photon distributions obtained with this protocol, always centered in a position proportional to the expectation value of the polarization of the detected photons and allowing to estimate the polarization expectation values always in agreement with the ones obtained with traditional PJ, all matching the theoretical expectations within the experimental uncertainties. These results shed important insight on the very foundations of quantum mechanics, especially in the long-standing debate about the ontic or epistemic nature of the wave-function, at the same time paving the way toward new quantum measurement methods with possible significant application for quantum technologies, and in particular, to quantum metrology [91, 92, 93], with the eventual possibility to exceed the quantum Cramer-Rao bound [94] thanks to the parameter dependence of the measurement procedure [95, 96]. Conceptualization, E.C., I.P.D., L.V., M.G. ([PERSON]) and M.G. ([PERSON]); methodology, A.A., E.C., F.P., I.P.D., L.V., M.G. ([PERSON]) and M.G. ([PERSON]); software, A.A., E.R., F.P., I.P.D., R.L. ; validation, I.P.D. and M.G. ([PERSON]); formal analysis, E.R. ; investigation, A.A., E.R., F.P. ; resources, A.T., F.V., R.L. ; data curation, F.P. ; writing--original draft preparation, all authors; writing--review and editing all authors; visualization, A.A., E.R. and F.P. ; supervision, I.P.D., G.B., M.G. ([PERSON]) and M.G. ([PERSON]); project administration, L.V. and M.G. ([PERSON]); funding acquisition, E.C., I.P.D., M.G. ([PERSON]) and L.V. All authors have read and agreed to the published version of the manuscript. Figure 5: Few-event photon counts distributions for the input state \(\left|\frac{17}{\theta 0}\pi\right\rangle\). In both figures, the first detection event is marked in yellow. While PJ **(a)** requires a measurement on an ensemble of identically prepared particles, PM **(b)** allows extracting the expectation value of our observable with just a single click. Yellow dashed line: \(x\) position corresponding to the theoretical expectation value of the polarization \(\left\langle A\right\rangle^{\text{th}}=-0.208\); red circles: FWHM of the corresponding distributions for the state \(\left|\frac{17}{\theta 0}\pi\right\rangle\) reported in Figure 4.  
**Funding:** This work has been financed by the projects 17 FUN01 'BeCOMe' and 17 FUN06 'SIQUST', both funded from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program. It has also been supported in part by the Israel Science Foundation Grant No. 2064/19 and the National Science Foundation--US-Israel Binational Science Foundation Grant No. 735/18. E.C. was supported by Grant No. FQXi-RFP-CPW-2006 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation, the Israel Innovation Authority under Projects No. 70002 and No.