## 3 Application to Classical Computing Device: Resonant Tunneling Diode
### Numerical Results  
In this subsection, to show how the quantum uncertainty of the values of the measured electrical currents can be controlled, we compute the autocorrelation function of the current \(I_{T}\) defined as:  
\[\langle I_{T}(t_{2})I_{T}(t_{1})\rangle=\int dI_{T}^{k}\int dI_{T}^{\omega}\;I _{T}^{k}I_{T}^{\omega}P(I_{T}^{k},I_{T}^{\omega}), \tag{9}\]  
where \(P(I_{T}^{k},I_{T}^{\omega})\) is the probability associated with the subsequent measurement of the multiparticle state \(|\Psi_{T}\rangle\) at two times \(t_{1}\) and \(t_{2}>t_{1}\), with \(I_{T}^{\omega}\) and \(I_{T}^{k}\) defined as the output values of the current at times \(t_{1}\) and \(t_{2}\), respectively. As indicated in the previous section, for practical reasons in the computation of the particle and displacement components of the current, we will use a wave function plus a Bohmian trajectory for each electron in the computation of the dispersion of \(I_{T}\) in the modified quantum device through expressions in Equations (7) and (8). The measurement of the current in an electrical device is a weak measurement process [8] in the sense that the perturbation of the wave function due to the measurement process is not very dramatic. This type of measurement is mathematically represented by three subsystems: the quantum system, the measuring apparatus plus an intermediate system or _ancilla_. In fact, the system interacts with the _ancilla_ during the measurement (not with the apparatus) and the apparatus measures the ancilla (not the system). This indirect way of getting information of the system by measuring the _ancilla_ implying that the output value of the total current has the quantum noise of the system plus the quantum noise of the _ancilla_. In our particular example, the _ancilla_ is just the cable (in fact there are a very large number of electrons) connecting the RTD with the ammeter [8]. As indicated in the previous section, for practical reasons in the computation of the particle and displacement components, in addition to the wave function, each electron will be described by a Bohmian trajectory. As we have mentioned, the elimination of the quantum uncertainty in the measured current implies that the total wave function in Equation (1) is an eigenstate of the many-particle operator in Equation (2). Then, the first measured value of the current \(I_{T}(t_{1})\) at time \(t_{1}\) has no influence on the output of a second measurement \(I_{T}(t_{2})\) at time \(t_{2}\), and it can be shown [17] that the two time probability of the autocorrelation given in Equation (9) can be written as the product of two independent probabilities, which leads to the following expression of the two time correlation in Equation (9),  
\[\langle I_{T}(t_{2})I_{T}(t_{1})\rangle = \int dI_{T}^{k}\int dI_{T}^{\omega}\ I_{T}^{k}I_{T}^{\omega}P(I_{T }^{k})P(I_{T}^{\omega}) \tag{10}\] \[= \int dI_{T}^{k}I_{T}^{k}P(I_{T}^{k})\int dI_{T}^{\omega}I_{T}^{ \omega}P(I_{T}^{\omega})\] (11) \[= \langle I_{T}(t_{2})\rangle\langle I_{T}(t_{1})\rangle\]  
The condition in Equation (11) is a test of the fact that the quantum uncertainty has disappeared in the measurement process of the modified device. Please note that from a pure engineering point of view, the autocorrelation mentioned above contains rich information on the high frequency response of the quantum device. In order to get the frequency response of the device we compute the power spectral density which is just the Fourier transform of the autocorrelation function (since the current signal has a constant mean, it is in a wide sense a stationary process, hence the auto-correlation depends only on the time difference \(t=t_{2}-t_{1}\). Therefore the auto-correlation function and the PSD form a Fourier transform pair).