### Theoretical Framework  
In the framework of quantum mechanics, given a quantum state \(|\psi\rangle\), we define the quantum expectation value of an observable \(A=\sum_{i}a_{i}|\phi_{i}\rangle\langle\phi_{i}|\) (with \(\sum_{i}|\phi_{i}\rangle\langle\phi_{i}|=\mathbb{I}\)) as the average of its eigenvalues \(a_{i}\) weighted on their respective probabilities \(p_{i}\):  
\[\langle A\rangle=\langle\psi|A|\psi\rangle=\sum_{i}p_{i}a_{i}\qquad\left(p_{i }=|\langle\psi|\phi_{i}\rangle|^{2}\right) \tag{1}\]  
Similarly to its classical counterpart, \(\langle A\rangle\) is understood as a statistical property.  
PM can be modeled as a standard von Neumann measurement [59] in which we couple the observable of interest \(A\) with a pointer \(P\) with a long and adiabatic interaction instead of the usual instantaneous one. Such interactions are mediated by the coupling \(g(t)\), which allows us to write the interaction Hamiltonian as  
\[\mathcal{H}_{int}=g(t)A\otimes P \tag{2}\]  
where the interaction intensity is \(g(t)=g/T\) for a time interval \(T\) and smoothly goes to zero before and after. If the coupling \(g(t)\) is smooth enough, we obtain the adiabatic limit, in which the state of the system \(|\psi\rangle\) does not change, thanks to the protection. In the limit \(T\rightarrow\infty\) and for bounded \(P\), one has \(\mathcal{H}_{int}\to 0\) while the state \(|\psi\rangle\) remains unchanged thanks to the preserving action of the protection mechanism. We can calculate, then, the shift of the energy of the eigenstate via first-order perturbation theory [3]:  
\[\delta E\simeq\langle\mathcal{H}_{int}\rangle=g\frac{\langle A\rangle P}{T} \tag{3}\]  
from which we can calculate the time evolution \(U\) associated to \(\mathcal{H}_{int}\) in the limit \(T\rightarrow\infty\):  
\[U\simeq\exp\biggl{(}-\frac{i}{\hbar}g\langle A\rangle P\biggr{)} \tag{4}\]  
resulting in a pointer wavefunction shift proportional to the expectation value \(\langle A\rangle\).  
A second protection scheme involves the so-called _active_ protection, based on the quantum Zeno effect [60], which consists of a series of repeated projections onto the initial state [61; 62; 63; 64; 65; 66; 67; 68; 69; 70] during the interaction described by the Hamiltonian in Equation (2), but in the non-adiabatic limit. In our experiment, we will focus on this quantum Zeno-type protection [6].  
From a quantum informational perspective, our experiment corresponds to a protocol in which [PERSON] produces a quantum state, which she transmits to [PERSON] together with the proper protection, implemented by [PERSON] as a black box when realizing the PM.  
The PM framework can be formalized with an equivalent description consisting of a series of \(K\) instantaneous weak interactions, described by a (weak) coupling constant \(g=\int_{t=0}^{T}g(t)dt\). Between two subsequent interactions, the active protection occurs, induced by the projector \(\Pi_{\psi}=|\psi\rangle\!\langle\psi|\). It is straightforward to show that, in the weak interaction approximation (\(g\ll 1\)), each of the \(K\) weak interaction/protection blocks evolves the system in the following way:  
\[|\psi\rangle\!\langle\psi|U|\psi\rangle\otimes|\varphi(x)\rangle=|\psi\rangle \!\langle\psi|\exp\!\left(-\frac{i}{\hbar}gA\otimes P\right)|\psi\rangle \otimes|\varphi(x)\rangle\approx|\psi\rangle\otimes|\varphi(x-g\langle\AA \rangle)\rangle \tag{5}\]  
where \(|\varphi(x)\rangle\) is the pointer initial wavefunction.  
From Equation (5), one can see how the PM induces a shift in the meter wavefunction, which is directly proportional to the expectation value \(\langle A\rangle\). PMs, then, allow us to directly estimate the expectation value \(\langle A\rangle\) for each single particle undergoing them, in sharp contrast with the concept of \(\langle A\rangle\) being only a statistical quantity.