## Appendix C Description of the Current Signal and Condition 2  
We discuss here which are the necessary considerations to design the modified (new) quantum device that allows us to assume that all electrons inside it have identical simultaneous contribution to the total current \(I_{T}\). The first step is identifying the proper single particle operator \(\hat{I}^{1}\). Then, once we identify \(\hat{I}^{1}\), we can compute the eigenstates \(|\psi^{1}(t_{in})\rangle\) and the eigenvalues \(I^{1}\) that will correspond to the measured output results. However, identifying the electrical current operator is not so simple for several reasons. First, the measured current in an ammeter at time \(t_{in}\) is not just the particle current, defined as the number of particles crossing a particular surface of the device, but it also includes the displacement current. The latter component of the electrical current is proportional to the time-dependent variations of the electric field on a particular surface of the device. Typically, such component is not relevant at low frequencies, but at larger frequencies no instantaneous current conservation at time \(t_{in}\) can be guaranteed without it. What is the operator associated with the measurement of the total, particle plus displacement, current? The answer is not trivial at all. In fact, the measurement of the electrical current in quantum electron devices has an additional difficulty. The measurement of the electrical current corresponds to a generalized or weak measurement, which are mathematically described by a POVM. So, the proper question is even more complicated now: What is the POVM associated with the measurement of the total, particle plus displacement, current? Fortunately, we can describe the measurement of the total electrical current in a quantum electron device without having to anticipate the POVM. We will use in this appendix an explanation of the measurement process of the total electrical current using the Bohmian quantum theory. Such theory is formally equivalent to the orthodox quantum theory, it gives the same empirical results, but it does not require to identify _a priori_ the measurement operator. Such theory defines a quantum system assigned to one electron by the orthodox wave function \(\psi^{1}(\vec{r},t)\) plus a quantum trajectory \(\vec{r}^{1}(t)=\{x^{1}(t),y^{1}(t),z^{1}(t)\}\) constructed from a velocity field given by the wave function itself. Such trajectory allows the definition of the properties of a quantum system, like the instantaneous electrical current \(I^{1}(t)\), independently of the fact of being measured or not. An identically prepared state for a second experiment will be described by the same wave function \(\psi^{2}(\vec{r},t)=\psi^{1}(\vec{r},t)\), but with a different trajectory \(\vec{r}^{2}(t)=\{x^{2}(t),y^{2}(t),z^{2}(t)\}\). The quantum uncertainty in the output value of the electrical current is due to the different initial positions of the trajectories, which describe an ensemble of identical experiments. The selection of the initial positions of the \(i\)-th trajectories is selected according to the quantum equilibrium [40]. According to the Bohmian theory, for the electron with trajectory \(\vec{r}^{1}(t)=\{x^{1}(t),y^{1}(t),z^{1}(t)\}\) and velocity \(\vec{v}^{1}(t)=\{v_{x}^{1}(t),v_{y}^{1}(t),v_{z}^{1}(t)\}\), the electrical (particle plus displacement) current \(I^{1}(t)\) generated in a surface of the quantum device is given by the Ramo-Shockley-Pellegrini theorem [41].