**1. Introduction**

Thanks to its accuracy and versatility, the Monte Carlo solution of the Boltzmann transport equation has been, for decades, the preferred computational tool to predict the DC, AC, transient, and noise performances of semi-classical electron devices [1]. In the past decade, however, due to the miniaturization of electronic devices (with active regions approaching the de Broglie wavelength of the transport electrons), a majority of the device modeling community has migrated from semi-classical to fully quantum simulation tools, marking the onset of a revolution in the community devoted to semiconductor device simulation. Today, a number of quantum electron transport simulators are available to the scientific community [2–4]. The amount of information that these simulators can provide, however, is mainly restricted to the stationary regime and therefore their predicting capabilities are still far from those of the traditional Monte Carlo solution of the semi-classical Boltzmann transport equation [1]. This limitation poses a serious problem in the near future as electron devices are foreseen to operate at the Terahertz (THz) regime. At these frequencies, the discrete nature of electrons in the active region is expected to generate unavoidable fluctuations of the current that could interfere with the correct operation of such devices both for analog and digital applications [5].

A formally correct approach to electron transport beyond the quasi-stationary regime lies on the description of the active region of an electron device as an open quantum system [6,7]. As such, one can then borrow any state-of-the-art mathematical tool developed to study open quantum systems [8,9]. A preferred technique has been the stochastic Schrödinger equation (SSE) approach [10–17]. Instead of directly solving equations of motion for the reduced density matrix, the SSE approach exploits the state vector nature of the so-called conditional states to alleviate some computational burden (and ensure a complete positive map by construction [18]). Even if this technique allows to always reconstruct the full density matrix, a discussion on whether dynamical information can be directly extracted from such conditional states in non-Markovian scenarios has appeared recently in the literature [19,20]. This debate is very relevant to us as we are interested in computing not only one-time expectation values (i.e., DC performance) but also dynamical properties (i.e., AC, transient, and noise), such as multi-time correlation functions, at THz frequencies. At these frequencies the environment correlations are expected to decay on a time-scale comparable to the time-scale relevant for the system evolution [21]. Furthermore, the displacement current becomes important at very high frequencies and a self-consistent solution of the Maxwell equations and the Schrödinger equation is necessary [21,22].

Some light on how to utilize the SSE technique to access dynamical information without the need of reconstructing the reduced density matrix has already been shed by Wiseman and Gambetta by acknowledging the Bohmian conditional wavefunction as the proper mathematical tool to describe general open quantum systems in non-Markovian scenarios [23,24]. In this work we reinforce this idea by showing that the Bohmian conditional wavefunction, together with the corresponding Bohmian trajectory, is an exact decomposition and recasting of the unitary time-evolution of a closed quantum system that yields a set of coupled, non-Hermitian, equations of motion that allows to describe the evolution of arbitrary subsets of the degrees of freedom on a formally exact level. Furthermore, since the measurement process is defined as a routine interaction between subsystems in Bohmian mechanics, conditional states can be used to describe either the measured or unmeasured dynamics of an open quantum system. As an example of the practical utility of the conditional wavefunctions, we present here a Monte Carlo simulation scheme to describe quantum electron transport in open systems that is valid both for Markovian or non-Markovian regimes and that guarantees a dynamical map that preserves complete positivity [25–29].

This paper is structured as follows. In Section 2 we provide a brief account on the SSE approach and on how nanoscale electron devices can be understood as open quantum systems. Section 3 focuses on the physical interpretation of the conditional states (i.e., system states conditioned on a particular value of the environment) in the contexts of the orthodox and Bohmian quantum mechanical theories. Section 4 provides an overall perspective on the points raised in the previous sections and puts into practice the conditional wavefunction concept to build a general purpose electron transport simulator, called BITLLES, beyond the steady state (Markovian) regime. As an example of the use of conditional states, numerical simulations of the THz current in a graphene electron device are presented in Section 5. Final comments and conclusions can be found in Section 6.

### **2. Electron Devices as Open Quantum Systems**

In this section we introduce the SSE approach to open quantum systems and discuss how it can be used to reconstruct the reduced density matrix. We then explain how a nanoscale electron device can be understood as an open quantum system and how the SSE approach can be applied to predict its performance.
