*2.2. Stochastic Schrodinger Equations*

A breakthrough in the computation of the reduced density matrix in Equation (2) came from the advent of the SSE approach [32]. The main advantage behind the SSE approach is that the unknown to be evaluated is in the form of a state vector (of *<sup>N</sup>*sys degrees of freedom) rather than a matrix (of size *N*2sys) and thus there is an important reduction of the associated computational cost. In addition, it provides equations of motion that, by construction, ensure a complete positive map [18] so that the SSE approach guarantees that the density matrix always yields a positive probability density, a requirement that is not generally satisfied by other approaches that are based on directly solving Equation (3) [33].

The central mathematical object in the SSE approach to open quantum systems is the conditional state of the system:

$$|\psi\_q(t)\rangle = \frac{\left(\langle q| \otimes \mathbb{I}\_{\rm sys}\rangle \,|\,\Psi(t)\rangle\right)}{\sqrt{P(q,t)}},\tag{5}$$

where *<sup>P</sup>*(*q*, *t*) = *ψq*(*t*)|*ψq*(*t*) = Ψ(*t*)|<sup>ˆ</sup>*I*sys ⊗ |*qq*| ⊗ ˆ*<sup>I</sup>*sys|Ψ(*t*) and |*q* are the eigenstates of the so-called unraveling observable *Q* ˆ belonging to the Hilbert space of the environment. To simplify the discussion, and unless indicated, *q* represents the collection of degrees of freedom of the environment. Using the eigenstates |*q* as a basis for the environment degrees of freedom, it is then easy to rewrite the full state |Ψ(*t*) as:

$$|\Psi(t)\rangle = \int dq \sqrt{P(q,t)} |q\rangle \otimes |\psi\_q(t)\rangle,\tag{6}$$

which can be simply understood as a Schmidt decomposition of a bipartite (open system plus environment) state. Thus, a complete set of conditional states can be always used to reproduce the reduced density matrix at any time as:

$$
\hat{\rho}\_{\text{sys}}(t) = \int dq P(q, t) |\psi\_{\emptyset}(t)\rangle\langle\psi\_{\emptyset}(t)|. \tag{7}
$$

Let us note that no specific (Markovian or non-Markovian) assumption was required to write Equation (7). In fact, the above definition of the reduced density matrix simply responds to the global unitary evolution in Equation (1), which (as depicted in Figure 1a) does not include the effect of any measuring apparatus.

### *2.3. Nanoscale Electron Devices as Open Quantum Systems*

At first sight, one could be inclined to say that a nanoscale electron device perfectly fits into the above definition of open quantum system. The open system would then be the device's active region and the environment (including the contacts, the cables, ammeter, etc.) the so called reservoirs or contacts (see Figure 1a). In addition, the observable of interest *A*ˆ in Equation (4) would be, most probably, the current operator ˆ*I*. As long as we are interested only in single-time expectation values, i.e., static or stationary properties, this picture (and the picture in Figure 1a) is perfectly valid. Therefore, the SSE approach introduced in Equations (5)–(7) can be easily adopted to simulate electron devices and hence to predict their static performance.

**Figure 1.** Panel (**a**): Schematic representation of an open quantum system, which can be partitioned into active region and environment. The evolution of the entire device is described by the state |Ψ(*t*) that evolves unitarily according to the time-dependent Schrödinger equation. Panel (**b**): Schematic representation of a measured open quantum system, which can be partitioned into meter, active region, and environment. The evolution of the device plus environment wavefunction is no longer unitary due to the (backaction) effect of the measuring apparatus.

However, if one aims at computing dynamical properties such as time-correlation functions, e.g., *I*(*t* + *<sup>τ</sup>*)*I*(*t*), then a valid question is whether such an expectation value is expected to be measurable at the laboratory. If so, what would then be the effect of the measurement of *I* at time *t* on the measurement of *I* at a later time *t* + *τ*?. Figure 1b schematically depicts this question by drawing explicitly the measuring apparatus (or meter). As it is well known, the action of measuring in quantum mechanics is not innocuous. It is quite the opposite: in many relevant situations, extracting information from a system at time *t* has a non-negligible effect on the subsequent evolution of the system and hence also on what is measured at a later time *t* + *τ*. Therefore, as soon as we are concerned about dynamic information (i.e., time-correlation functions), we need to ask ourselves whether an approach to open quantum systems such as the SSE approach can be of any help. In the next section we will answer this question and understand whether the conditional states |*ψq*(*t*) defined in Equation (5) do include the backaction of the measuring apparatus depicted in Figure 1b.

### **3. Interpretation of Conditional States in Open Quantum Systems**

The conditional states in Equation (5) were first interpreted as a simple numerical tool [32], that is, exploiting the result in Equation (7) as a numerical recipe to evaluate any expectation value of interest. This interpretation is linked to the assumption that the operator *A*ˆ in Equation (4) is the physically relevant operator (associated to a real measuring apparatus), while the operator *Q*ˆ associated to the definition of the conditional state in Equation (5) is only a mathematical object with no attached

physical reality, i.e., it merely represents a basis. In more recent times, however, it has been generally accepted that the conditional states in Equation (5) can be interpreted as the states of the system conditioned on a type of sequential (sometimes referred to as continuous) measurement [34] of the operator *Q*ˆ of the environment (now representing a physical measuring apparatus that substitutes the no longer needed operator *A*ˆ) [6,12,35]. From a practical point of view, this last interpretation is very attractive as it would allow to link the conditional states, |*ψq*(*t*), at different times and thus compute time-correlation functions without the need of introducing the measuring apparatus or of reconstructing the full density matrix. Whether or not this later interpretation is physically sound in general circumstances is the focus of our discussion in the next subsections.

### *3.1. The Orthodox Interpretation of Conditional States*

Let us start by discussing, in the orthodox quantum mechanics theory, what is the physical meaning of the conditional states that appear in Equation (5). When the full closed system follows the unitary evolution of Figure 1a, the conditional state |*ψq*(*t*) can be understood as the (renormalized) state that the system is left in after projectively measuring the property *Q* of the environment (with outcome *q*). This can be easily seen by noting that the superposition in Equation (6) is, after a projective measurement of *Q*, reduced (or collapsed) to the product state

$$<|\Psi\_q(t)\rangle = \sqrt{P(q,t)}|q\rangle \otimes |\psi\_q(t)\rangle. \tag{8}$$

It is important to notice that the conditional state |*ψq*(*<sup>t</sup>* ) at a later time, *t* > *t*, can be equivalently defined as the state of the system when the superposition in Equation (6) is measured at time *t* and yields the outcome *q*. This interpretation, however, is only valid if no previous measurement (in particular at *t*) has been performed, as depicted in Figure 2a. Otherwise, the collapse of the wavefunction at time *t*, yielding the state *<sup>P</sup>*(*q*, *<sup>t</sup>*)|*q*⊗| *ψq*(*t*), should be taken into account in the future evolution of the system, which would not be the same as if the measurement had not been performed at the previous time. Therefore, the equation of motion of the conditional states, as defined in Equation (5), cannot be, in general, the result of a sequential measurement protocol such as the one depicted in Figures 1b or 2b. This conclusion seems obvious if one recalls that our starting point was Figure 1a, where there is no measurement.

### 3.1.1. Orthodox Conditional States in Markovian Scenarios

Even if the conditional states solution of the SSE cannot be generally interpreted as the result of a sequential measurement, such an interpretation has been proven to be very useful in practice for scenarios that fulfill some specific type of Markovian conditions. We are aware that there is still some controversy on how to properly define Markovianity in the quantum regime (see, e.g., Ref. [18]), so it is our goal here only to acknowledge the existence of some regimes (i.e., particular observation time intervals) of interest where the role of the measurement of the environment has no observable effects. In this regime, Figure 1a,b as well as Figure 2a,b can be thought to be equivalent.

**Figure 2.** Panel (**a**): Schematic representation of the SSE approach. The states of the system conditioned on a particular value of the environment at time *t*, |*ψq*(*t*), can be given a physical meaning only if no measurement has been performed at a previous time *t* < *t*. This approach can be always used to reconstruct the correct reduced density matrix of the system at any time but cannot be used to link in time the conditional states for non-Markovian scenarios. Panel (**b**): Schematic representation of a sequential measurement. The wavefunction of the system plus environment is measured sequentially. In this picture, the link between the states of the full system plus environment at different times is physically motivated.

In our pragmatical definition of Markovianity the entanglement between system and environment decays in a time scale *tD* that is much smaller than the observation time interval *τ*, i.e., *tD τ*. In this regime, the environment itself can be thought of as a type of measuring operator (as appears in generalized quantum measurement theory [36]) that keeps the open system in a pure state after the measurement. The open system can be then seen as an SSE in which the stochastic variable *qt* (sampled from the distribution *<sup>P</sup>*(*qt*, *t*)) is directly the output of a sequential measurement of the environment. The stochastic trajectory of this conditioned system state generated by the (Markovian) SSE is often referred to as a quantum trajectory [6,12,35] and can be used, for example, to evaluate time-correlation functions of the environment as:

$$
\langle Q(t)Q(t+\tau)\rangle \stackrel{t\_D \ll \tau}{=} \int \int P(q\_t, t)P(q\_{t+\tau}, t+\tau)q\_t q\_{t+\tau} dq\_t dq\_{t+\tau} = \langle Q(t)\rangle \langle Q(t+\tau)\rangle.\tag{9}
$$

Let us emphasize that the stochastic variables *qt* and *qt*<sup>+</sup>*τ* in Equation (9) are sampled, separately, from the probability distributions *<sup>P</sup>*(*qt*, *t*) = *ψq*(*t*)|*ψq*(*t*) and *<sup>P</sup>*(*qt*<sup>+</sup>*τ*, *t* + *τ*) = *ψq*(*<sup>t</sup>* + *<sup>τ</sup>*)|*ψq*(*<sup>t</sup>* + *<sup>τ</sup>*). Therefore, as we have schematically depicted in Figure 3, no matter how the trajectories {*qt*} are connected in time, one always obtains the correct time-correlation function *Q*(*t*)*Q*(*t* + *<sup>τ</sup>*).

**Figure 3.** Schematic representation of the combined system plus environment wavefunction (blue Gaussians) measured at different times that result in a state of the system |*ψq*(*t*) conditioned to the set of environment values {*qt*} shown in dark blue circles. In the Markovian regime there exists no specific recipe about how the different *qt*'s must be connected in time (colored solid lines). No matter how these points are connected in time, one always gets the right expectation value in Equation (9).

It is important to realize that we started our discussion on the physical meaning of the Markovian SSE with an open system whose environment is not being measured (see Figures 1a and 2a). Noticeably, we have ended up discussing an environment that is being measured at every time interval *τ* (see Figure 2b). How is that possible? Well, the reason is that measuring the environment at time *t* does not affect the system conditional states at a later time *τ* when the built-in correlations in the environment due to the measurement at time *t* decay in a time interval *tD* much smaller than the time interval between measurements *τ*. In other words, Figure 1a,b as well as Figure 2a,b are not distinguishable when *tD τ*. In this sense, the Markovian regime has some similarities with a classical system, where it is accepted that information can be extracted without perturbation.

### 3.1.2. Orthodox Conditional States in Non-Markovian Scenarios

For nanoscale devices operating at very high (THz) frequencies, the relevant dynamics and hence the observation time interval *τ* are both below the picoseconds time-scale and the previous assumption of Markovianity, i.e., *tD τ*, starts to break down. Under the condition *tD* ∼ *τ*, non-Markovian SSE have been proposed which allow an alternative procedure for solving the reduced state |*ψq*(*t*) [17,33,37–41]. However, non-Markovian SSEs constructed from Equation (5), unlike the Markovian SSEs, suffer from interpretation issues [17]. In the non-Markovian regime, the perturbation of the environment due to the quantum backaction of a measurement at time *t* would not be washed out in the time lapse *τ* ∼ *tD* and hence the joint probability distribution would not become separable, i.e., *<sup>P</sup>*(*qt*, *qt*<sup>+</sup>*τ*)) = *<sup>P</sup>*(*qt*)*P*(*qt*<sup>+</sup>*τ*). As a direct consequence, connecting in time the different solutions *qt* and *qt*<sup>+</sup>*τ* (sampled independently from the probability distributions *<sup>P</sup>*(*qt*, *t*) and *<sup>P</sup>*(*qt*<sup>+</sup>*τ*, *t* + *τ*) as in Figure 3 to make a trajectory "would be a fiction" [17,19,20]. Here, the word "fiction" means that the time-correlations computed in Equation (9) are wrong, i.e., the expectation value in Equation (9) would simply be different from the experimental result.

According to D'Espagnat the above discussion can be rephrased in terms of the so-called proper and improper mixtures [42]. Following D'Espagnat arguments, the reduced density matrix in Equation (7) is an improper mixture because it has been constructed by tracing out the degrees of freedom of the environment. On the contrary, a proper mixture is a density matrix constructed to simultaneously define several experiments where a closed system is described, at each experiment, by different pure states. Due to our ignorance, we do not know which pure state corresponds to which experiment, so we only know the probabilities of finding a given pure state. D'Espagnat argues that the ignorance interpretation of the proper density matrix, cannot be applied in the improper density matrix discussed here (See Appendix A). To understand why under a Markovian regime open systems can be described by pure states (using a proper mixture), we remind that Markovianity implies conditions on the observation time. For a given correlation time *tD*, a given open system can be in the Markovian or non-Markovian regimes depending on the time of observation *τ*. That is, for small enough observation times all open systems are non-Markovian and hence must be understood as an improper mixture. On the contrary, for large enough observation times, open systems can behave as closed systems (with a negligible entanglement with the environment) and be effectively represented by pure states.

### *3.2. The Bohmian Interpretation of Conditional States*

So, under non-Markovian (i.e., the most general) conditions, the conditional states |*ψq*(*t*) can be used to reconstruct the reduced density matrix as in Equation (7) but cannot be used to provide further information on its own. This interpretation problem is rooted in the fact that orthodox quantum mechanics only provides reality to objects whose properties (such as *q*) are being directly measured. However, as explained in the previous subsection, it is precisely the fact of introducing the measurement of *q* (without including the pertinent backaction on the system evolution) which prevents the conditional states |*ψq*(*t*) of the non-Markovian SSE from being connected in time for *tD* ∼ *τ*. In this context, a valid question regarding the interpretation of |*ψq*(*t*) is whether or not we can obtain information of, e.g., the observable *Q* without perturbing the state of the system. The answer given by orthodox quantum mechanics is crystal clear: this is not possible (except for Markovian conditions) because information requires a measurement, and the measurement induces a perturbation. Notice, however, that the assumption that only measured properties are real is not something forced on us by experimental facts, but it is a deliberate choice of the orthodox quantum theory. Therefore, we here turn to a nonorthodox approach: the Bohmian interpretation of quantum mechanics [43–48].

A fundamental aspect of the Bohmian theory is that reality (of the properties) of quantum objects does not depend on the measurement. That is, the values of some observables, e.g., the value of the positions of the particles of the environment, exist independently of the measurement. If *q* is the collective degree of freedom of the position of the particles of the environment and *x* is the collective degree of freedom of the position of particles of the system; then, the Bohmian theory defines an experiment in the laboratory by means of two basic elements: (i) the wavefunction *q*, *x*|Ψ(*t*) = <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*) and (ii) an ensemble of trajectories *Qi* (*t*), *X<sup>i</sup>* (*t*) of the environment and of the system. We use a superindex *i* to denote that each time an experiment is repeated, with the same preparation for the wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*), the initial positions of the environment and system particles can be different. They are selected according to the probability distribution |Ψ(*X<sup>i</sup>* , *Q<sup>i</sup>*, 0)| 2 [44]. The equation of motion for the wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*) is the time-dependent Schrödinger equation in Equation (1), while the equations of motion for the environment and system trajectories *Qi* (*t*), *X<sup>i</sup>* (*t*) are obtained by time-integrating the velocity fields *vq*(*<sup>x</sup>*, *q*, *t*) = *Jq*(*<sup>x</sup>*, *q*, *<sup>t</sup>*)/|Ψ(*<sup>x</sup>*, *q*, *t*)| 2 and *vx*(*<sup>x</sup>*, *q*, *t*) = *Jx*(*<sup>x</sup>*, *q*, *<sup>t</sup>*)/|Ψ(*<sup>x</sup>*, *q*, *t*)| 2 respectively. Here, *Jq*(*<sup>x</sup>*, *q*, *t*) and *Jx*(*<sup>x</sup>*, *q*, *t*) are the standard current densities of the environment and the system respectively. We highlight the (nonlocal) dependence of the Bohmian velocities of the particles of the environment on the particles of the system, and vice-versa. This shows

just the entanglement between environment and system at the level of the Bohmian trajectories. According to the continuity equation

$$\frac{d|\Psi(\mathbf{x},q,t)|^2}{dt} + \nabla\_{\mathbf{x}}(v\_{\mathbf{x}}(\mathbf{x},q,t)|\Psi(\mathbf{x},q,t)|^2) + \nabla\_{\theta}(v\_{\theta}(\mathbf{x},q,t)|\Psi(\mathbf{x},q,t)|^2) = 0,\tag{10}$$

the ensemble of trajectories {*Q*(*t*), *X*(*t*)} = {*Q*<sup>1</sup>(*t*), *<sup>X</sup>*<sup>1</sup>(*t*), *Q*<sup>2</sup>(*t*), *<sup>X</sup>*<sup>2</sup>(*t*)....*Q<sup>M</sup>*(*t*), *X<sup>M</sup>*(*t*)} with *M* → ∞ can be used to reproduce the probability distribution |Ψ(*<sup>x</sup>*, *q*, *t*)|<sup>2</sup> at any time. Thus, by construction, the computation of ensemble values from the orthodox and Bohmian theories are fully equivalent, at the empirical level.

From the full wavefunction *<sup>x</sup>*, *q*|Ψ(*t*) = <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*) (solution of Equation (1)) and the trajectories *Q<sup>i</sup>*(*t*), *<sup>X</sup><sup>i</sup>*(*t*), one can then easily construct the Bohmian conditional wavefunction of the system and environment as *ψ* ˜ *<sup>Q</sup><sup>i</sup>*(*t*)(*<sup>x</sup>*, *t*) = <sup>Ψ</sup>(*<sup>x</sup>*, *Q<sup>i</sup>*(*t*), *t*), and *<sup>ψ</sup>*˜*X<sup>i</sup>*(*t*)(*q*, *t*) = <sup>Ψ</sup>(*X<sup>i</sup>*(*t*), *q*, *t*) respectively. Notice that this Bohmian definition of conditional states does not require to specify if the system is measured or not because the ontological nature of the trajectories {*Q*(*t*), *X*(*t*)} does not depend on the measurement. Consequently, the conditional wavefunctions *<sup>ψ</sup>*˜*Q<sup>i</sup>*(*t*)(*<sup>x</sup>*, *t*), with the corresponding Bohmian trajectories, contain all the required information to evaluate dynamical properties of the system no matter whether Markovian or non-Markovian conditions are being considered. This can be seen by noticing that the velocity of the trajectory *X<sup>i</sup>*(*t*) given by *vq*(*X<sup>i</sup>*(*t*), *Q<sup>i</sup>*(*t*)) can be equivalently computed either from (the *<sup>x</sup>*−spatial derivatives of) the global wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *Q*, *t*) evaluated at *X<sup>i</sup>*(*t*) and *Q<sup>i</sup>*(*t*) or from (the *x*-spatial derivative of) the conditional wavefunction *<sup>ψ</sup>*˜*Q<sup>i</sup>*(*t*)(*<sup>x</sup>*, *t*) evaluated at *<sup>X</sup><sup>i</sup>*(*t*). In other words, the Bohmian velocities computed from <sup>Ψ</sup>(*<sup>x</sup>*, *Q*, *t*) or *<sup>ψ</sup>*˜*Q<sup>i</sup>*(*t*)(*<sup>x</sup>*, *t*) are identical. Thus, in a particular experiment *i* and for a given time *t*, the dynamics of the Bohmian trajectory *X<sup>i</sup>*(*t*) can be computed either from *<sup>ψ</sup>*˜*Q<sup>i</sup>*(*t*)(*<sup>x</sup>*, *t*) or from <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*).

The Bohmian conditional wavefunction of the system can now be connected to the orthodox conditional wavefunction in Equation (5) by imposing *Q<sup>i</sup>*(*t*) ≡ *qt*. Then one can readily write:

$$|\tilde{\psi}\_{q\_l}(t)\rangle = P(q\_{t\prime}t)|\psi\_{q\_l}(t)\rangle. \tag{11}$$

At first sight, one can think that the difference between the Bohmian and orthodox conditional states is just a simple renormalization constant *<sup>P</sup>*(*qt*, *t*) (see Appendix B for a more detailed explanation of the role of this renormalization constant). However, the identity in Equation (11) has to be understood as to be satisfied at any time *t*, which implies that the following identity should prevail:

$$Q^i(t) \equiv q\_{t\prime} \qquad \forall \; t \tag{12}$$

We emphasize the importance of Equation (12) in ensuring the accomplishment of Equation (11). If we consider another experiment *Qj*(*t*) ≡ *qt* , we have to define another conditional state |*ψ*˜*qt*(*t*). It can happen that, at a particular time *t* ≡ *t*1, both conditional states become identical i.e., |*ψ*˜*qt*1 (*<sup>t</sup>*1) = <sup>|</sup>*ψ*˜*qt*1 (*<sup>t</sup>*1). However, this does not imply that both conditional wavefunctions can identically be used in the computation of time-correlations. This is because every Bohmian trajectory has a fundamental role in describing the history of the Bohmian conditional state for one particular experiment. Therefore, the trajectory *Q<sup>i</sup>*(*t*) uniquely describes the evolution of the conditional wavefunction |*ψ*˜*qt*(*t*) for one experiment (labeled by the index *i* in the Bohmian language) the same way as the trajectory *Qj*(*t*) and the conditional wave function |*ψ*˜*qt*(*t*) describes the experiment labeled by *j*. As we said, |*ψ*˜*qt*1 (*<sup>t</sup>*1) = <sup>|</sup>*ψ*˜*qt*1 (*<sup>t</sup>*1) are the same orthodox conditional states, but do not necessarily represent the same Bohmian conditional wavefunction. This subtle difference explains why SSEs cannot be connected in time and used to study the time-correlation of non-Markovian open system whereas the same can be done through the Bohmian conditional states, without any ambiguity.

The mathematical definition of the measurement process in Bohmian mechanics and in the orthodox quantum mechanics differs substantially [44]. In the orthodox theory a collapse (or reduction) law, different from the Schrödinger equation, is necessary to describe the measurement process [45]. Contrarily, in Bohmian mechanics the measurement is treated as any other interaction as far as the degrees of freedom of the measuring apparatus are taken into account [44]. Therefore, while in the orthodox theory the conditional states |*ψqt*(*t*) cannot be understood without the perturbation of the full wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*), in Bohmian mechanics the states |*ψ*˜*qt*(*t*) do have a physical meaning even when the full wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*) is unaffected by the measurement of the environment [23]. Interestingly, this introduces the possibility of defining what we call "unmeasured (Bohmian) conditional states" when it is assumed that there is no measurement or that the measurement of *qt* at time *t* has a negligible influence on the subsequent evolution of the conditional state.

Importantly, the Bohmian conditional states and the corresponding Bohmian trajectories can be used not only to reconstruct the reduced density matrix in Equation (7) at any time but the environment trajectories {*Q*(*t*)} allow us to correctly predict any dynamic property of interest including time-correlation functions, e.g.,

$$
\langle Q(t)Q(t+\tau)\rangle = \frac{1}{M} \sum\_{i=1}^{M} Q^i(t)Q^i(t+\tau) = \int \int P(q\_t, q\_{t+\tau}) q\_t q\_{t+\tau} dq\_t dq\_{t+\tau} \tag{13}
$$

where *M* → ∞ is the number of experiments (Bohmian trajectories) considered in the ensemble and we have defined *<sup>P</sup>*(*qt*, *qt*<sup>+</sup>*τ*) = 1*M* ∑*Mi*=<sup>1</sup> *<sup>δ</sup>*(*qt* − *Q<sup>i</sup>*(*t*))*δ*(*qt*<sup>+</sup>*τ* − *Q<sup>i</sup>*(*t* + *<sup>τ</sup>*)). As it is shown in Figure 4, the evaluation of Equation (13) and any other dynamic property when *tD* ∼ *τ* can be done only by connecting the (Bohmian) trajectories at different times in accordance with the continuity equation in Equation (10). This is in contrast with the evaluation of the dynamics in the Markovian regime where any position of the environment at time *t*1 can be connected to another position of the environment at time *t*2 (see Figure 3) and hence we can write *Q*(*t*)*Q*(*t* + *τ*) *tDτ* = 1*M*<sup>2</sup> <sup>∑</sup>*Mi*,*<sup>j</sup> Q<sup>i</sup>*(*t*)*Qj*(*t* + *<sup>τ</sup>*). This very relevant point was first explained by Gambetta and Wiseman [23,24].

**Figure 4.** Schematic representation of the combined system+environment wavefunction (blue Gaussians) that is measured at different times and results in a Bohmian conditional state |*ψ*˜*q*(*t*) conditioned to the set of environment values {*qt*} shown in dark blue circles. In the non-Markovian regime only those values from the set of values satisfying the continuity equation in Equation (10) can be linked in time to form a trajectory (shown as connected red circles). Dashed lines represent connections that do not follow the continuity equation and hence cannot be used to evaluate any dynamic property.

Although the Bohmian theory can also provide measured properties of the system that coincide with the orthodox results in Figure 2b, let us emphasize once more the merit of the unmeasured properties provided by the Bohmian theory, which remains mainly unnoticed in the literature. As it has been already explained, in the orthodox theory, measuring a particular value of the environment property *q* at time *t* cannot be conceived without the accompanying perturbation of the wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *q*, *t*). Under non-Markovian conditions, it is precisely this perturbation that prevents the conditional states of the system |*ψqt*(*t*) from being connected in time to form a trajectory. Contrarily, in Bohmian mechanics, the existence of the environment trajectories {*Q*(*t*)}, even in the absence of any measurement, allows the possibility of connecting in time the conditional states |*ψ*˜*qt*(*t*) even when *tD* ∼ *τ*.

Note that in the Bohmian framework, where the measurement apparatus is simply represented by an additional number of degrees of freedom interacting with the system (i.e., without requiring any additional collapse law), a discussion about measured and unmeasured properties of quantum systems is pertinent [49]. At a practical level, the measurement of many classical systems implies non-negligible perturbations. In particular, electronic devices at high frequencies are paradigmatic examples where such perturbations occur. It is well-known that the experimental setup (for e.g., a coaxial cable) connecting the electronic device to the meter induces dramatic perturbations in high-frequency measurements. An important task for device engineers is to determine what part of the measured signal is due to the intrinsic behaviour of the electron device and what part is due to rest of the experimental setup. When trying to predict the "intrinsic" behaviour of the electronic devices, the coaxial cables are modelled by "parasitic" capacitors or inductors to account for their "spurious" effect. Even the measurement of the whole experimental setup is repeated twice, with and without the "intrinsic" device under test (DUT), to subtract the results and determine experimentally the "intrinsic" properties of the electronic device alone. Such "intrinsic" properties of the electronic devices are what we define in this manuscript as the unmeasured properties of quantum systems.

### **4. Bohmian Conditional Wavefunction Approach to Quantum Electron Transport**

The different notions of reality invoked by the orthodox quantum theory and Bohmian mechanics lead to practical differences in the abilities that these theories can offer to provide information about quantum dynamics. Specifically, we have shown that contrarily to orthodox quantum mechanics, Bohmian mechanics allows to physically interpret (i.e., link in time) the conditional states of the SSE approach in general non-Markovian scenarios. The reason is that whereas in the Bohmian theory the reality of the current is independent of any measurement, the orthodox theory gives reality to the electrical current only when it is being measured (this is the so-called eigenstate–eigenvalue link). From the practical point of view, this has a remarkable consequence. In the Bohmain approach the total current can be defined in terms of the dynamics of the electrons (Bohmian) trajectories without the need to define a measurement operator. As it will be shown in this section, the possibility of computing the total current at high frequencies without specifying the measurement operator is certainly a grea<sup>t</sup> advantage of the Bohmian approach in front of the orthodox one [44]. In particular, one can then avoid cumbersome questions like, is the measurement operator of the electrical current strong or weak? If weak, how weak? How often do such operator acts on the system? Every picosecond, every femtosecond? At high frequencies, how we introduce the contribution of the displacement current in the electrical current operator?

In this section we provide a brief summary of the path that the authors of this work followed for developing an electron transport simulator based on the use of Bohmian conditional states. The resulting computational tool is called BITLLES [28,29,50–56]. Let us start by considering an arbitrary quantum system. The whole system, including the open system, the environment, and the measuring apparatus, is described by a Hilbert space H that can be decomposed as H = H*x* ⊗ H*q* where H*x* is the Hilbert space of the open system and H*q* the Hilbert space of the environment. If needed, the Hamiltonian H*q* can include also the degrees of freedom of the measuring apparatus as

explained in Section 3.2. We define *x* = {*<sup>x</sup>*1, *<sup>x</sup>*2...*xn*} as the degrees of freedom of *n* electrons in the open system, while *q* collectively defines the degrees of freedom of the environment (and possibly the measuring apparatus). The open system plus environment Hamiltonian can then be written as:

$$
\hat{H} = \hat{H}\_{\mathbf{q}} \otimes \mathbb{I}\_{\mathbf{x}} + \mathbb{I}\_{\mathbf{q}} \otimes \hat{H}\_{\mathbf{x}} + \hat{\mathcal{V}} \tag{14}
$$

where *H*ˆ x is the Hamiltonian of the system, *H*ˆ *q* is the Hamiltonian of the environment (including the apparatus if required), and *V*ˆ is the interaction Hamiltonian between the system and the environment. We note at this point that the number of electrons *n* in the open system can change in time and so the size of the Hilbert spaces H*x* and H*q* can depend on time too.

The equation of motion for the Bohmian conditional states *x*|*ψ*˜*qt*(*t*) = *ψ*˜*qt*(*<sup>x</sup>*, *t*) in the position representation of the system can be derived by projecting the many-body (system-environment) Schrödinger equation into a particular trajectory of the environment *qt* ≡ *Q*(*t*), i.e., [26,57]:

$$i\hbar \frac{d\tilde{\varphi}\_{q\_t}(\mathbf{x}, t)}{dt} = \langle q\_t | \otimes \langle \mathbf{x} | \hat{H} | \Psi(t) \rangle + i\hbar \nabla\_q \langle q | \otimes \langle \mathbf{x} | \Psi(t) \rangle \Big|\_{q = q\_t} \frac{dq\_t}{dt}.\tag{15}$$

Equation (15) can be rewritten as:

$$i\hbar\frac{d\bar{\psi}\_{\rm lt}(\mathbf{x},t)}{dt} = \left[-\frac{\hbar^2}{2m}\nabla\_x^2 + \mathcal{U}\_{q\_l}^{eff}(\mathbf{x},t)\right]\tilde{\psi}\_{\rm lt}(\mathbf{x},t),\tag{16}$$

where

$$i\bar{\Pi}\_{q\_t}^{eff}(\mathbf{x},t) = \mathbb{U}(\mathbf{x},t) + V(\mathbf{x},q\_t,t) + \mathcal{A}(\mathbf{x},q\_t,t) + i\mathcal{B}(\mathbf{x},q\_t,t). \tag{17}$$

In Equation (17), *<sup>U</sup>*(*<sup>x</sup>*, *t*) is an external potential acting only on the system degrees of freedom, *<sup>V</sup>*(*<sup>x</sup>*, *qt*, *t*) is the Coulomb potential between particles of the system and the environment evaluated at a given trajectory of the environment, A(*<sup>x</sup>*, *qt*, *t*) = −*h*¯ 2 2*m* ∇<sup>2</sup> *q*<sup>Ψ</sup>(*<sup>x</sup>*, *q*, *<sup>t</sup>*)/Ψ(*<sup>x</sup>*, *q*, *t*) *q*=*qt* and B(*<sup>x</sup>*, *qt*, *t*) = *h*¯ <sup>∇</sup>*q*<sup>Ψ</sup>(*<sup>x</sup>*, *q*, *<sup>t</sup>*)/Ψ(*<sup>x</sup>*, *q*, *t*) *q*=*qt q*˙*t* (with *q*˙*t* = *dqt*/*dt*) are responsible for mediating the so-called kinetic and advective correlations between system and environment [26,57]. Equation (16) is non-linear and describes a non-unitary evolution.

In summary, Bohmian conditional states can be used to exactly decompose the unitary time-evolution of a closed quantum system in terms of a set of coupled, non-Hermitian, equations of motion [26,57–59]. An approximate solution of Equation (16) can always be achieved by making an educated guess for the terms A and B according to the problem at hand. Specifically, in the BITLLES simulator the first and second terms in Equation (17) are evaluated through the solution of the Poisson equation [29]. The third and fourth terms are modeled by a proper injection model [60] as well as proper boundary conditions [56,61] that include the correlations between active region and reservoirs. Electron-phonon decoherence effects can be also effectively included in Equation (16) [25].

In an electron device, the number of electrons contributing to the electrical current are mainly those in the active region of the device. This number fluctuates as there are electrons entering and leaving the active region. This creation and destruction of electrons leads to an abrupt change in the degrees of freedom of the many body wavefunction which cannot be treated with a Schrödinger-like equation for *ψ*˜*qt*(*<sup>x</sup>*, *t*) with a fixed number of degrees of freedom. In the Bohmian conditional approach, this problem can be circumvented by decomposing the system conditional wavefunction *ψ*˜*qt*(*<sup>x</sup>*, *t*) into a set of conditional wavefunctions for each electron. More specifically, for each electron *xi*, we define a single particle conditional wavefunction ˜ *ψ*˜*qt*(*xi*, *<sup>X</sup>*¯*i*(*t*), *t*), where *<sup>X</sup>*¯*i*(*t*) = {*<sup>X</sup>*1(*t*), .., *xi*−<sup>1</sup>(*t*), *xi*+1, .., *Xn*(*t*)} are the Bohmian positions of all electrons in the active region except *xi*, and the second tilde denotes the single-electron conditional decomposition that we have considered on top of the conditional

decomposition of the system-environment wavefunction. The set of equations of motion of the resulting *n*(*t*) single-electron conditional wavefunctions inside the active region can be written as:

$$i\hbar \frac{d\tilde{\vec{\psi}}\_{\Psi\_{\rm l}}(\mathbf{x}\_{1}, \vec{X}\_{1}(t), t)}{dt} = \left[ -\frac{\hbar^{2}}{2m} \nabla\_{\mathbf{x}\_{1}}^{2} + \tilde{\mathcal{U}}\_{\Psi\_{\rm l}}^{eff}(\mathbf{x}\_{1}, \vec{X}\_{1}(t), t) \right] \tilde{\vec{\psi}}\_{\Psi\_{\rm l}}(\mathbf{x}\_{1}, \vec{X}\_{1}(t), t) \tag{18}$$

$$i\hbar \frac{d\tilde{\psi}\_{\rm q}(\mathbf{x}\_{n}, \vec{\mathbf{X}}\_{n}(t), t)}{dt} = \left[ -\frac{\hbar^{2}}{2m} \nabla\_{\mathbf{x}\_{n}}^{2} + \tilde{\mathcal{U}}\_{\rm q}^{eff}(\mathbf{x}\_{n}, \vec{\mathbf{X}}\_{n}(t), t) \right] \tilde{\psi}\_{\rm q}(\mathbf{x}\_{n}, \vec{\mathbf{X}}\_{n}(t), t). \tag{19}$$

That is, the first conditional process is over the environment degrees of freedom and the second conditional process is over the rest of electrons within the (open) system.

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We remind here that, as shown in Figure 2b, the active region of an electron device (acting as the open system) is connected to the ammeter (that acts as the measuring apparatus) by a macroscopic cable (that represents the environment). The electrical current provided by the ammeter is then the relevant observable that we are interested in. Thus, the evaluation of the electrical current seems to require keeping track of all the degrees of freedom, i.e., of the system and the environment, which is of course a formidable computational task (see (d) Table 1). At THz frequencies, however, the electrical current is not only the particle current but also the displacement current. It is well-known that the total current defined as the particle current plus the displacement current is a divergence-less vector [21,22]. Consequently, the total current evaluated at the end of the active region is equal to the total current evaluated at the cables. So the variable of the environment associated to the total current, *qt* ≡ *<sup>I</sup>*(*t*), can be equivalently computed at the borders of the open system. The reader is referred to Ref. [62] for a discussion on how *I*(*t*) can be defined in terms of Bohmian trajectories with the help of a quantum version of the Ramo–Schokley–Pellegrini theorem [63]. In particular, it can be shown that the total (particle plus displacement) current in a two-terminal devices can be written as [63]:

$$I(t) = \frac{\varepsilon}{L} \sum\_{i=1}^{n(t)} v\_{x\_i}(X\_i(t), \mathcal{X}\_i(t), t) = \frac{\varepsilon}{L} \sum\_{i=1}^{n(t)} \text{Im} \left( \frac{\nabla\_{x\_i} \tilde{\Psi}\_{q\_l}(\mathbf{x}\_i, \mathcal{X}\_i(t), t)}{\tilde{\Psi}\_{q\_l}(\mathbf{x}\_i, \mathcal{X}\_i(t), t)} \right) \Big|\_{\substack{x\_i = X\_i(t)}},\tag{20}$$

where *L* is the distance between the two (metallic) contacts, *e* is the electron charge, and *vxi* (*Xi*(*t*), *<sup>X</sup>*¯*i*(*t*), *t*) is the Bohmian velocity of the *i*-th electron inside the active region. Let us note that *I*(*t*) is the electrical current given by the ammeter (although computed by the electrons inside the open system). Since the cable has macroscopic dimensions, it can be shown that the measured current at the cables is just equal to the unmeasured current (taking into account only the simulation of electrons inside the active region) plus a source of (nearly white) noise which is only relevant at very high frequencies [62]. The basic argumen<sup>t</sup> is that the (non-simulated) electrons in the metallic cables have a very short screening time. In other words, the electric field generated by an electron in the cable spatially decreases very rapidly due to the presence of many other mobile charge carriers in the cable that screen it out. Thus, the contribution of this outer electron to the displacement current at the border of the active region is negligible [64].

Summarizing, for the computation of the current at THz frequencies, the degrees of freedom of the environment can be neglected without any appreciable deviation from the correct current value [62]. This introduces an enormous computational simplification as shown (e) in Table 1. This is, for the specific scenarios that we are interested in, the computation cost of the Bohmian conditional wavefunction approach has the same computational cost as the orthodox SSE approach (see Table 1). Yet, in contrast to the orthodox conditional states, which can be used only to evaluate the dynamics of quantum systems in the Markovian regime, the Bohmian conditional states provide direct information on the dynamics of both Markovian or non-Markovian systems.


**Table 1.** An estimation of the computational cost (in memory) of different approaches mentioned in the text. Here *Nsys* and *Nenv* are the number of degrees of freedom of the system and the environment while *M* denotes the number of elements required.
