Remarks on time-dependent [current]-density functional theory for open quantum systems

Time-dependent [current]-density functional theory for open quantum systems (OQS) has emerged as a formalism that can incorporate dissipative eﬀects in the dynamics of many-body quantum systems. Here, we review and clarify some formal aspects of these theories that have been recently questioned in the literature. In particular, we provide theoretical support for the following conclusions: (1) contrary to what we and others had stated before, within the master equation framework, there is in fact a one-to-one mapping between vector potentials and current densities for fixed initial state, particle– particle interaction, and memory kernel; (2) regardless of the first conclusion, all of our recently suggested Kohn–Sham (KS) schemes to reproduce the current and particle densities of the original OQS, and in particular, the use of a KS closed driven system, remains formally valid; (3) the Lindblad master equation maintains the positivity of the density matrix regardless of the time-dependence of the Hamiltonian or the dissipation operators; (4) within the stochastic Schro¨dinger equation picture, a one-to-one mapping from stochastic vector potential to stochastic current density for individual trajectories has not been proven so far, except in the case where the vector potential is the same for every member of the ensemble, in which case, it reduces to the Lindblad master equation picture; (5) master equations may violate certain desired properties of the density matrix, such as positivity, but they remain as one of the most useful constructs to study OQS when the environment is not easily incorporated explicitly in the calculation. The conclusions support our previous work as formally rigorous, offer new insights into it, and provide a common ground to discuss related theories.

There is also an analogous theory due to Vignale, 8 named timedependent current-density functional theory (TDCDFT), which establishes a one-to-one correspondence between vector potentials and current densities, and guarantees the existence of a noninteracting KS system which, evolving under a fictitious KS vector potential, reproduces the current density of the original system.Since the particle density is related to the current density via the continuity equation, the original particle density is also reproduced in such KS system.
The performance of TDDFT depends on the energy functionals one employs and the ability to express the desirable observables in terms of the particle density.However, its role as a practical method to obtain dynamical properties of manybody systems is so far unrivaled by other ab initio methods due to the accuracy it provides with the currently available functionals versus the computational cost it entails.General prescriptions to compute excited state energies and their oscillator strengths via TDDFT have been reported, [9][10][11][12][13][14][15] and have been successfully used to study a wide variety of phenomena, such as chemical [16][17][18][19][20][21] and nonlinear optical [22][23][24] properties in nanomaterials, resonant energy transfer, 25,26 and many-body effects in solid state systems, [27][28][29] and lifetimes of atomic resonance states, 30,31 among many others.7][48][49][50][51][52][53][54][55] We hereby restrict the definition of OQS to the domain of systems that exchange energy-but not particles, with an environment.1][62][63] In our work on OQS-TDDFT, we have established precise conditions for -V and -A representability for the evolution of general ME. 51,53,93An interesting outcome of our investigation is that particle and current densities of OQS can be reproduced in closed driven systems (CDS) evolving unitarily.Recently, in an unrefereed posting in the arXiV e-Print server 64 and its sequel, 65 D'Agosta and Di Ventra (DADV) have casted doubts on several of our results, potentially generating awareness but also confusion among the practitioners of the field, since it is not clear whether they or we are correct.We regard this occasion as a good opportunity to present what we believe to be an objective account of the subject.The goal of this article is to clarify our work 51,53,93 in comparison with theirs 48,49,66 in the broad context of OQS in TDDFT.The paper is structured as follows: in Section 1, we establish the notation which will be used throughout the paper, and in Section 2, we address a series of formal issues of TDDFT for OQS which have been a potential source of confusion in the literature.This work paves the theoretical foundations for further development of practical methods to simulate dynamics of many-body OQS.

I Notation
Consider the general evolution of the density matrix in the form of a ME: Here r(t) is the density matrix describing the quantum state of the system of interest, Kðt; t 0 Þ is a memory kernel which describes how the state at times 0 r t 0 r t affect the dynamics at time t.If Kðt; t 0 Þ / dðt À t 0 Þ, the ME is Markovian; otherwise, it is non-Markovian.T ðtÞ is an inhomogeneous term that encodes initial system-bath correlations.The discussion of this article does not pertain T ðtÞ, so hereafter we shall set it to zero for simplicity.The many-body Hamiltonian of the system is given by: where we are working in the gauge where the scalar potential vanishes.ri and pi are the position and momentum of the i-th particle in the system, -A is the external potential, U is the interparticle interaction, e and m are the charge and mass of the particles.We shall be interested in the particle and current density operators, nðr Þ ¼ P i d r À ri , ĵðr; tÞ ¼ 1 2 , where hi t indicates a trace with respect to r(t).We emphasize the fact that the time dependence of hO ˆ(r, t)i t will stem both from the explicit time dependence of the operator O ˆ(r, t) and from the evolution of r(t) due to the ME (1).The property of interest can also be a non-observable arbitrary functional of r(t), such as the purity O( r, t) = P t = Tr(r(t) 2 ).Let us define some functions under fixed initial density matrix r(0), interparticle potential U( r i , r j ), and memory kernel Kðt; t 0 Þ (see Fig. 1): 1. F: -A( r, t)r(t) maps vector potentials to density matrices via the ME of eqn (1), with fixed r(0), U( r i , r j ), and Kðt; t 0 Þ.Note that G = I n,j F, G ˜= I j F. In one of our articles (Yuen-Zhou, Rodrı ´guez-Rosario, and Aspuru-Guzik, 51 hereafter denoted as YRA), we already defined the first three functions and showed in its Theorem 1 an analog of the RG and Vignale theorems, namely, that G is a one-to-one map.This statement is still correct, despite the challenges to it by ref. 64 In practice, this would be achieved through a non-interacting KS system aimed at reproducing these two observables.†

II Remarks and discussions
A. G ˜is injective (one-to-one) too We proceed to answer the following question: is G ˜also injective?If so, analogous conclusions to the ones above associated with G would hold too.In YRA, we made an incorrect claim, implying that the answer is ''no.''Here, we correct our original statement.THEOREM.G ˜is a one-to-one map.PROOF.The proof is similar to the one for Theorem 1 presented in YRA, provided some subtleties are addressed.Consider two systems: the original one, associated to r(t), which evolves according to eqn (1).The auxiliary one, associated with r 0 (t), which evolves under the primed equation of motion, and yield the same current density as the original system: Here, the ''primed'' Hamiltonian Ĥ0 S ðtÞ is the same as eqn (2) except for the substitution A -A 0 .Assume throughout this proof that r(0) = r 0 (0), U = U 0 , and Kðt; t 0 Þ ¼ K 0 ðt; t 0 Þ for all t À t 0 Z 0. Suppose that for all r A R 3 and t Z 0, the current densities are the same in both systems.We ask the question: under these circumstances, can Ãa Ã0 for some value of r and t?The answer will be ''no'', hence proving the theorem.
The equations of motion for the particle and current densities of the original system are: Here, lðr; tÞ ¼ Tr nðrÞ R t 0 dt 0 Kðt; is the leakage term that restores the continuity equation for OQS. 51Whereas a closed system always satisfies the continuity equation, an OQS may not satisfy the continuity equation in general when the leakage term is nonzero.8][69] Nevertheless, for purposes of this article, we shall operationally keep referring to ĵðr; tÞ D E t as the current density.
The rest of the terms are given by Dðr; tÞ . Explanations of these terms can be found in YRA.Analogous primed relations hold for the auxiliary system.Let us Taylor expand eqn (5) about t = 0 as Oðr; tÞ . Equating powers of t, we obtain, after collecting t lÀ1 terms.Note that eqn (4) can also be written as: for all 0 r k o N. If Kðt; t 0 Þ is a smooth function of the time arguments, l k depends at most on @ kÀ1 Kðt; t 0 Þ @t kÀ1 t 0 ¼t and @ kÀ1 r @t kÀ1 (see YRA).
-A couples directly to the system and indirectly to the bath through the system.If there are any memory effects at time t associated with -A, they must be due to -A( r, t 0 ) for t 0 strictly smaller than t.In other words, -A couples to the system, the system couples to the bath, and only afterwards, in a second order process in the system-bath interaction can this information return to the system as memory through Kðt; t 0 Þ.By these considerations, @ kÀ1 Kðt; t 0 Þ @t kÀ1 t¼t 0 can depend at most on @ kÀ1 r @t kÀ1 also depends at most on ) through the Hamiltonian in eqn (2).Therefore, in the case of Kðt; t 0 Þ smooth as a function of t and t 0 , l k ( r ) depends at most on ).A proof using formal arguments derived from projector operator methods yield the same conclusion. 70,71For the important Markovian limit where Kðt; t 0 Þ is not a smooth function and Kðt; t 0 Þ ¼ LðtÞdðt À t 0 Þ for some superoperator L. By the same arguments as above, @ k L @t k should depend at most on common situation is when L is constant in time).l k can also depend on @ k r @t k , which depends at most on ÃkÀ1 ðr Þ.Therefore, the summary is that in either case, l k depends at most on r ).The next step in the proof is an induction.Consider the strong inductive hypothesis: for all 0 r k o l where l Z 1. Two statements follow: (a) Eqn ( 8) together with the arguments in the previous paragraph imply that: ‡ which means that at each step, we prove that the particle densities in both systems must be the same.(b) Let us Taylor expand eqn (6) and its primed version, and use eqn ( 8) and (10) to collect terms associated with t l : where DA = A 0 À A. Eqn (11) is the same as eqn (10) in YRA.The same immediate consequence of that investigation is obtained, namely, that the right hand of the equation contains D -A l as its highest order derivative (we refer the reader to YRA for a detailed derivation of this conclusion).Hence, due to the hypothesis in eqn (9), the right hand side vanishes identically to zero.
We now prove eqn (9) for the basis case for the induction, l = 1.Since the initial states in both systems are the same, n 0 ðr Þ ¼ n 0 0 ðr Þ ¼ hnðr Þi 0 .Eqn (8) holds for any k, in particular for k = 0.This yields: Eqn ( 11) also holds for l = 0, and gives D Ã1 ðr Þ ¼ 0, proving the basis case.By induction, ) for all t in the domain of convergence of all the Taylor expansions.The proof can be repeated Taylor expanding about different times until they cover the entire t Z 0 domain.
In passing, note that eqn (4) together with the rest of the hypotheses imply eqn (10), which means hnðr Þi t ¼ hnðr Þi 0 t once the induction proof is over.In YRA, we had regarded the latter as an additional hypothesis, but here we realize that such assumption is a consequence of the other hypotheses of the current theorem.
Discussion. .In other words, our YRA scheme was redundant, but correct.In particular, our central result (Theorem 3 in YRA) of using a CDS as a KS scheme holds, and its variant in the context of TDDFT 53 has been correctly proposed, together with a practical Markovian bath functional.
The TDDFT quantum Drude approach for dissipation due to Neuhauser 50 can also be regarded as a mapping from an original OQS to a CDS KS scheme.Also, restating the ideas from last section, let us illustrate how G is injective, but not necessarily bijective.Simply, take a h j( r, t)i t given by some r(t), which in turn, is produced by an To aid in the following arguments, we reproduce Theorem 3 from YRA: ''Theorem 3. (A,C Kohn Sham scheme) There exists an auxiliary closed KS system which starting in the state r KS (0) evolves unitarily under a KS Hamiltonian H ˆKS (t): producing particle and filtered current densities that are related to the particle and current densities of the original many body interacting OQS governed by (1)  C are not unique (in fact, they do not need to be for Theorem 3 in YRA to be true).As explained in the proof in YRA, the continuity equation of the unitary evolution of the KS system is given by: @hnðr Þi KS;t @t ¼ À rÁ ĵfilt ðr; tÞ D E

KS
but this is not unique.Here, the vector field points in the x direction, r = (x, y, z), and the lower limit of the integral is at (0, y, z).The arbitrariness of -C( r, t), however, is immaterial: so long as it satisfies eqn (15), it will subtract the correct amount of leakage current from the total KS current, so that the second constraint in eqn ( 14) is fullfilled.
The proof for the theorem in discussion (see YRA) explains how to construct . Furthermore, we can construct Ã00 KS from another -C 00 which also guarantees eqn (14).In Hence, the ''total'' KS vector potential A KS and C00 þ Ã00 KS will be different, in consistency with Theorem 1, which reduces to Vignale's theorem as the KS system is taken to be closed. 8

C. Positivity of the Kossakowski-Lindblad master equation irrespective of time-dependence of Hamiltonian and dissipation operators
A quantum ME is the generic denomination of an equation of motion for the density matrix of an OQS.][73][74] Alternatively, phenomenological approaches are also often advocated.
A popular form of Markovian ME which is often used in the literature is the generalized Kossakowski-Lindblad (KL) equation: [75][76][77] @r @t ¼ LðrÞ Ài Ĥ where both H ˆand L ˆi can be time-dependent.This ME is known to preserve the positivity and trace of the density matrix under its evolution, irrespective of the time-dependence of H ˆand L ˆi.To verify this, we expand r(t + Dt) for small Dt: Using eqn (18), we can derive the so-called Kraus operators M i (Dt), which are: for i > 0, and An important property of the Kraus operators is that they satisfy: as can be easily checked from eqn (20) and (21).Notice that, in general, M i (Dt) are time dependent if L ˆi are too.Let us write rðtÞ ¼ P i p j x j ðtÞ x j ðtÞ , where {|x i (t)i} is the basis that diagonalizes r(t) with p j Z 0 for all j.Notice that up to O(Dt 2 ): which evidences that r(t + Dt) is positive semidefinite if r(t) is.The preservation of the trace can also be readily shown using eqn (22)  and the cyclic invariance of the trace: The proofs above are a ''backwards'' adaptation of the standard textbook derivation of the KL equation as the generator of a completely positive map.See, for instance, the textbook. 78or completeness, we now introduce the concept of a semigroup.Consider the integrated form of the equation of motion for r(t) in the form of a dynamical map, r(t) = F t,0 r(0), where F t,0 is a dynamical map that propagates the density matrix from 0 to the final time t.The semigroup property can be expressed as the following identity for the composition map: 71 This journal is c the Owner Societies 2013 Discussion.On the one hand, as it is stated in ref. 49, the semigroup property will in general not be satisfied in general for time dependent H ˆor L ˆi.For instance, these operators at times belonging to the interval [t, t + s] can differ significantly from their values at [0, s].On the other hand, for the case where both types of operators are time-independent, it has been shown 76 that L is the most general form of the generator of a quantum dynamical semigroup, with the dynamical map being the exponential map: F s;0 ¼ e Ls .However, contrary to what 49 and DADV claim, neither positivity nor trace-preservation depend on the time-dependence of H ˆor L ˆi, as we have explicitly shown here.

D. No proof yet for the Runge-Gross theorem analog for individual trajectories in the stochastic Schro ¨dinger equation
It is well known that if H ˆdoes not depend on the state of the system, eqn (18) can be ''unraveled'' as the evolution of an ensemble of Stochastic Schro ¨dinger Equations (SSE) which reconstructs the density matrix upon appropriate manipulation of the calculated trajectories. 60,79Hence, for this situation, the SSE yields the same density matrix dynamics as the KL-ME in eqn (18).There are, however, several interesting features of the SSE that make it attractive compared to the ME, such as its lower numerical cost 80 as well as the novel conceptual insights it provides based on measurement theory. 81In any case, the theorems in the work by Burke and coworkers 46,47 and by us in YRA and, 53 all of which are derived from the ME approach, can be implemented in the SSE KS scheme.An important point is that the KS vector or scalar potential would need to be calculated as a functional of the ensemble-averaged ĵðr; tÞ D E or hn ˆ(r, t)i, respectively.Hence, one could devise a parallel algorithm where all the trajectories in the ensemble are propagated simultaneously, and at each time-step, the ensemble-averaged ĵðr; tÞ D E or hn ˆ(r, t)i computed, from which the KS potentials would be calculated as functionals.However, this is not what 49 propose.Instead, they suggest the evolution of ensembles of wavefunctions evolving under different stochastic KS potentials which depend on each individual trajectory (see ref. 48, 49 and 64 for their general discussion, and ref. 66 for their TDDFT implementation).Unfortunately, the one-to-one mapping from single trajectory stochastic external potentials to the corresponding stochastic particle and current densities has not been proven.Hence, insofar this proof does not exist, the KS scheme in ref.48    r, t) acts on each member of the ensemble, which coincides with the domain where the SSE is equivalent to the ME.Another verification is that our equation of motion for the current 51,53 reduces to theirs in the limit where the memory kernel is of the KL form.A proof relying on these equations of motion cannot be a proof for the Runge-Gross analog for individual trajectories in the Stochastic Schro ¨dinger Equation.At this point, we are not in the position to claim that there is no such Runge-Gross analog, but so far, if such proof exists, it has not been published.
It seems that authors in ref.48 and 49 are aware of this issue but they still advocate their method.According to them, having a single -A KS for the entire ensemble ''requires that the exchange-correlation vector potential included also the statistical correlations of the direct Coulomb interaction at different points in space'' (see discussion around eqn (43) in ref. 49).It is difficult for us to understand what this statement precisely means, and why the problem would be solved with the method they suggest.Also in the cited page, the authors worry about lack of positivity of the density matrix by using only one timedependent -A KS potential for all the trajectories.As shown in the previous section, this preoccupation is unfounded.
Having pointed out these subtleties, it might be the case that even if the KS scheme of ref.48 and 49 is not rigorous, it is a pragmatic and useful approximate computational tool.However, it would be interesting to obtain further justification of the preference for their method as opposed to the more rigorous one described above.
We emphasize that at this point we are not claiming there is no Runge-Gross analogue theorem for the correspondence between stochastic densities and stochastic potentials for each trajectory.There simply has not been a published proof for it yet.Clearly, regardless of whether this proof appears, our ME approaches remain valid.

E. Deficiencies of arbitrary master equation approaches
A ME can be systematically derived from a microscopic model under the Nakajima-Zwanzing projector operator method. 71,74his approach is by far the most general one for OQS, and applies both to Markovian and non-Markovian dynamics, but might suffer from some problems due to the approximations involved in their derivations, such as the lack of positivity.3][84] An alternative approach is the restriction of the equations of motion to satisfy some minimal requirements such as positivity and trace preservation.The KL equation is an example of the latter approach, 85,86 but non-Markovian analogues also exist. 87Even in these cases, it is difficult to construct MEs which satisfy all the desirable requirements of an OQS evolution.For instance, Kohen and Tannor have shown that no Markovian ME simultaneously satisfies the desired triad of positivity, translational invariance, and asymptotic approach to thermal equilibrium. 88iscussion.DADV criticize our study of generalized ME because they might not maintain positivity in general.As implied above, we are not aware of a ME which fullfills all the desirable physical requirements of an OQS evolution.This is true even in the context of the KL equation, which although does not violate positivity, is not translationally invariant, yielding different dissipative dynamics under translation of the coordinate system, a phenomenon which is not physical. 88][91][92] Even if ME suffer from unavoidable deficiencies, they are one of the most promising ways to study OQS coupled to large environments.Hence, they are an invaluable theoretical tool and their connection with TDDFT should pave the way for the description of a wide variety of interesting OQS.

III Conclusions
In this article, we have addressed several formal issues concerning our previous work 51,53 and that of others 48,49,66 in the topic TDDFT for OQS.In summary, we have found an incorrect speculation we had previously advocated in ref. 51, namely, that the map G ˜from vector potentials to current densities is not injective, when in reality it is, as the theorem in this article establishes.Fortunately, the theorem in this article does not render any of the theorems in YRA as invalid, and therefore, all the suggested KS schemes proposed in our investigation remain rigorously applicable.In particular, we have clarified that using a KS CDS to compute the particle and current densities of the original OQS is well founded.The intuition behind this result is that the OQS under consideration exchange energy with an environment, but not particles.Time-dependent potentials driving closed systems also pump energy in and out of the system.Since only simple variables of the OQS such as particle and current densities profiles are supposed to be mimicked by the KS system, it is not surprising that we can achieve this with a CDS.Obviously, there are many properties that will not be the same in both systems, but that is expected from the TD(C)DFT theorems, as they only claim that those properties may be calculated as the functionals of particle and current densities, which in principle, are extracted correctly from the KS system.We also argued that uniqueness of several KS potentials is irrelevant in our scheme.
Contrary some recent views in the literature, we have shown that the KL equation maintains positivity even in the case of time-dependent Hamiltonians or dissipation operators.We also carefully compared the ME approach against the SSE formalism advocated by ref. 48, 49 and 66.If the potential does not depend on the stochastic wavefunction itself, then the SSE is equivalent to the KL-ME.However, the cited works place an important emphasis to the case where the potential depends on the stochastic wavefunction, yielding an ensemble of different potentials that govern each of the stochastic trajectories.Under such circumstances, just as those authors claim, no closed equation of motion for the observables can be derived.Yet, since their TDDFT proof in ref.48 depends on such equations of motion, they must limit the practice of their theory to the same realm as the one of the MEs, unless there is a pragmatic reason to skip the rigor they have provided themselves.An alternative is to attempt to prove a Runge-Gross analog for stochastic trajectories, which to our knowledge, has not been published so far.MEs may violate certain desirable physical properties, but they are still very important constructs for the study of OQS.As far as our work is concerned, our mappings of TD(C)DFT based on MEs provide a rigorous framework to develop computational tools for the study of OQS.The development of dissipative functionals is at its infancy, 50,53,55 but we foresee interesting work on this realm.
In summary, in this article, we have attempted to clarify several formal aspects of TDDFT for OQS, emphasizing on certain mappings and KS schemes which have caused some confusion in the recent literature.We have also compared the different schemes based on ME and SSE.We hope that the results and arguments presented here will aid the further development of the field.

Fig. 1
Fig.1Relevant mappings for TDCDFT.Given a fixed initial state r(0), interparticle potential U(r i , rj ), and memory kernel Kðt; t 0 Þ, F maps vector potentials Ã(r, t) to density matrices r(t) via the evolution of the ME.Once r(t) is calculated, any properties of the system can be calculated from it, yielding the maps I O from density matrices to arbitrary properties O ˆ(r, t), and in particular, I n,j to the particle and current density pair nðrÞ h i t ; ĵðr; tÞ D E t , and I j to the current density ĵðr; tÞ D E t only.G = I n,j F, G ˜= I j F are the composition maps from Ã(r, t) directly to nðrÞ h i t ; ĵðr; tÞ D E t and ĵðr; tÞ D E t, respectively.In YRA, we have shown that G is one-to-one, yielding a well-defined inverse G À1 whose domain is the image set of G.In the current article, we show that G ˜is also one-to-one, and that the inverse G ˜À1 exists, with its domain being the image set of G ˜.

-G
the variable Gðr; tÞ À e C r; t m hnðr Þi KS;t .By virtue of the Helmholtz decomposition, it is always possible to write -J , where > and J denote the transverse and longitudinal components of the vector field, defined by the relations rÁ -G > = 0 and r Â -G J = 0.The divergenceless condition of -G > implies that it may be expressed as the curl of a vector field - 8,48,51 Theorem 1 in YRA and the Theorem in this article altogether claim that both G and G ˜are injective maps.
t in the KS system because then, the properties of the original system can be expressed as Oðr; tÞ ¼ I O F GÀ1 ĵðr; tÞ D E t .In YRA, we had criticized the KS scheme presented in ref.48 and 49 based on the fact that, in general, it is not possible to reproduce hn ˆ(r )i t and ĵðr; tÞ D E t simultaneously using a KS system which is restricted to dissipate with the same bath operators as the original system.The latter is indeed the case for microscopically derived MEs of the Lindblad form in the weak system-bath coupling regime, where the dissipation operators are in general non-local, yielding finite leakage terms l( t .A correction to this statement by the same authors is later reported in ref. 64 and 65.It is important, however, to stress that none of the KS schemes we suggest in YRA, which are intended to reproduce both hn ˆ(r )i t and ĵðr; tÞ D E t via alternative methods, are affected by the theorem in the present article, and remain valid as KS This journal is c the Owner Societies 2013 64 and Kðt; t 0 Þ.Hence, the domain of G À1 is restricted to the image set of G, as explained before and in ref.51.Contrary to previous criticism,64this restriction allows for the well-defined inversion of observables as functionals of hnðrÞi r i , r j t ; ĵðr; tÞ D E t via Oðr; tÞ ¼ I O FG À1 hnðrÞi t ; ĵðr; tÞ D E t .B. No inconsistency in the mapping from the original open quantum system to the Kohn-Sham closed driven system DFT schemes are about.Since the OQS under consideration exchange energy with an environment, but do not lose particles as time evolves, it is quite conceivable that there exists a CDS which mimics the particle and current density profiles of the OQS by pumping energy in and out of the system via the time-dependent external potentials.Obviously, there will be properties which differ in both systems, such as the purity, P KS (t)).We shall call-A KS ( r, t) the KS vector potential, and -C( r, t) the leakage potential.''Discussion.Theorem 3 in YRA establishes the possibility to reproduce both hn ˆ(r )i t and ĵðr; tÞ D E t as expectation values of appropriately defined operators using a CDS ðKðt; t 0 Þ ¼ 0Þ.Clearly, due to the leakage term l( t = Tr(r(t) 2 ) a Tr(r KS (t) 2 ) = P t,KS .Whereas P t changes in time as it becomes entangled with the bath, P t,KS stays constant throughout the unitary evolution.Nevertheless, there is no flaw in the logic of the formalism we have presented: even P t could in principle be calculated as a functional of the current density, which is obtained through the KS system, P t ¼ I O F GÀ1 ĵðr; tÞ D E t , where O = P t .This functional, however, might not be trivial to find in practice.Note that -A KS and - not unique, and two different total KS vector potentials yield the same particle and filtered current , respectively.As opposed to what is claimed in ref. 64, this fact is not in contradiction with Theorem 1.In general, the total current densities ĵðr; tÞ D E KS;t due to -C + - equation of motion for the current density cannot be derived.Hence, the equations of motion for the current displayed in ref.48 and 49do not hold for this case.They hold for the case where the same -A(-