## Appendix A Generalization to an Unmodified Quantum Device with Many Electrons  
In the text, to simplify the notation and the discussion of our protocol, we have assumed that the unmodified (original) quantum device has only one electron in the active region.This assumption is obviously unrealistic in many scenarios for either classical or quantum computing. For example, in quantum computing, we need a nanoscale devices with, at least, the number of electrons equal to the number of qubits we want to deal with. We show in this appendix that the very same protocol described in the text for one electron can be straightforwardly generalized to deal with an unmodified (original) quantum device with many electrons present in the active region of the nanoscale device. We consider an unmodified (original) quantum device with \(M\) electrons in the active region. To simplify the discussion, we will write the quantum state in the position representation. Then, each electron is described by the degree of freedom \(x_{j}\) with \(j=1,2,...,M\). Such quantum system is described by the \(M\)-particle wave function:  
\[\psi(x_{1},x_{2},...,x_{M},t_{in}) \tag{1}\]  
Notice that we do not assume any particular shape of this \(M\)-particle wave function so that exchange and Coulomb interaction among the \(M\) electrons is taken into account in the definition of this \(M\)-particle system (without any restriction). Now, we consider \(N\) set of \(M\) electrons which are prepared under the same conditions as the ones in Equation (1). In other words, the quantum dynamics of each set of these \(M\) electrons can be described by the same wave function in Equation (1). Thus, we define \(N\times M\) degrees of freedom for the modified quantum device as \(x_{j}^{i}\) with \(i=1,2,...,N\) counting the repetitions and \(j=1,2,...,M\) counting the number of electrons in each repetition. We define \(N\) wave function identical to Equation (1), but with a superindex \(i=1,2,...,N\) indicating which is the repetition we are dealing with:  
\[\psi^{i}(x_{1}^{i},x_{2}^{i},...,x_{M}^{i},t_{in}) \tag{2}\]  
Introducing the wave function in Equation (2) into the quantum state of the modified quantum device written in Equation (1), in the position representation, we have:  
\[\Psi_{T}(x_{1}^{1},...x_{M}^{N},t_{in})=\psi^{1}(x_{1}^{1},...,x_{M}^{1},t_{ in})\cdot\psi^{2}(x_{1}^{2},...,x_{M}^{2},t_{in})\cdot....\cdot\psi^{N}(x_{1}^{N },...,x_{M}^{N},t_{in}) \tag{3}\]  
Clearly, we have assumed in our definition of \(\Psi_{T}(x_{1}^{1},...x_{M}^{N},t_{in})\) in Equation (3) that there is no Coulomb or exchange interaction between the subset of electrons \(\{x_{1}^{k},...,x_{M}^{k}\}\) (for a treatment of the effect of exchange interaction see Appendix D) and the subset of electrons \(\{x_{1}^{l},...,x_{M}^{l}\}\) for any \(j\
eq k\), but no restriction is imposed on the interaction among the \(M\) electrons of each subset. The rest of the demonstration till the final result in Equation (6) with \(\sigma_{I_{T}}^{2}=0\) are basically the same that we wrote in Equations (3)-(5) for the evaluation of mean values. A relevant point in our discussion is that even with our best technological means to exactly reproduce the same wave function in Equation (1) with another set of \(M\) electrons, the quantum dynamics of these new set of electrons is not exactly identical to the previous one because of the inherent quantum uncertainty. The key element in our demonstration is that all these \(N\) different uncertainties of the set of \(M\) electrons have to follow an identical probability distribution given by the same wave function \(\psi\) in Equation (1). We notice that we have discussed in this appendix that if a system of \(N\times M\) electrons are described by Equation (11) (what we call **condition 1** in the text) and if we use a (center-of-mass-type) operator in Equation (2) (what we called **condition 2** in the text), then, the uncertainty in the measurement disappears in the \(\lim N\rightarrow\infty\).