II. QUANTUM MONTE-CARLO METHOD Our starting point is a system (S) interacting with a surrounding environment (E). We assume here that the total system (S+E) is described by the Hamiltonian H$_{S}$ (resp. H$_{E}$ ) acts on the system (resp. env.) only while V induces a coupling between the two sub-systems. Starting from an initial total density D (0), the dynamical evolution is given by the Liouville von-Neumann equation on the density: In many physical situations, the total number of degrees of freedom to follow in time prevents from solving exactly this equation. One of the leitmotiv of Open Quantum System (OQS) theory is to find accurate approximations for the system evolution without following explicitly irrel- evant degrees of freedom associated to the environment and therefore reduces the complexity of the initial prob- lem. Conventional strategy to treat dissipation and fluc- tuation in an open quantum system is to reduce the in- formation to the system density only ρ$_{S}$ ( t ) = Tr E ( D ( t )) while accounting approximately for environment effect. Here, we use a different strategy, the dynamics of the total system is first replaced by a set of stochastic evo- lutions where the total density remains separable along each path, i.e. D = ρ$_{S}$ ( t ) ⊗ ρ$_{E}$ ( t ). Then, the stochastic evolution of the environment is projected onto the rele- vant degrees of freedom to obtain a closed equation for the system density. It is shown that the new approach provides a proper treatment of dissipation and fluctua- tion for a system coupled to a surrounding heat-bath. Recently, new stochastic formulations [34, 35, 42, 43] have been developed to study the system+environment problem that avoid evaluation of non-local memory ker- nels although non-Markovian effects are accounted for exactly (see also [35, 36, 42, 44–47]). One example of such a theory based on quantum Monte-Carlo method is presented here. Hereafter, it is assumed that the coupling is separable: V = Q ⊗ B where Q and B act on the system and en- vironment respectively. For simplicity, initial separable density is considered, i.e. D (0) = ρ$_{S}$ (0) ⊗ ρ$_{E}$ (0). We want to replace the evolution of the total density (Eq. (2)) by an ensemble of stochastic evolutions of both the system and environment such that: The discussion above for one time step can then be iterated to show that the exact dynamics of a sys- tem+environment could be replaced by an average over an ensemble of stochastic evolutions of separable densi- ties [34–36]. To be really useful, mainly two difficulties should be overcome (i) in general, the environment cor- responds to a large number of degrees of freedom that could not be followed explicitly in time. (ii) the numeri- cal implementation of such a theory is possible only if the statistical errors do not growth too fast during the time- evolution. This statistical errors are directly connected to the number of trajectories necessary to accurately de- scribe the physical process. Difficulty (i) is solved in the next section by projecting out the effect of the environ- ment on the system leading to a closed equation for the system density only. Let us first concentrate on statis- tical errors. At any time, a measure of the statistical fluctuation around the average trajectory is given by the average evolution over the separable densities that evolve according to Eq. (3) identify with the exact Li- ouville von Neumann equation of motion (Eq. (2)). The possibility to use simple Gaussian noises to incorporate the environment effect might appear surprising. Indeed, noises used in standard approaches for Open Quantum Systems generally reflect properties of the environment. It should be however kept in mind that such environment dependent noises appear once the environment dynam- ics has been projected out on the system density evo- lution. Anticipating the discussion of section II B, once such a projection has been made, the Gaussian noises in- troduced here, transform into new random variables that explicitly depend on the environment properties. Therefore, under the condition where the overline denotes the stochastic average. Along each path, the total density remains separable, i.e. D ( t ) = ρ$_{S}$ ( t ) ⊗ ρ$_{E}$ ( t ). Starting from such a density, at time t + dt , the average evolution deduced from Eq. (3) is given by where dξ$_{S/E}$ and dλ$_{S/E}$ denote Markovian Gaussian stochastic variables with zero means and where we use Ito convention of stochastic calculus [48]. In the follow- ing, we assume in addition that 2 dξ$_{S}$dλ$_{E}$ = dλ$_{S}$dξ$_{E}$ = 0 , (4) H = H$_{S}$ + H$_{E}$ + V. (1) i $_{ℏ}$dD ( t ) dt = [ H, D ( t )] . (2) dD ( t ) = dt i $_{ℏ}$[ h$_{S}$ + h$_{E}$, D ( t )] + dξ$_{S}$dξ$_{E}$ ( Q ⊗ B ) D ( t ) + dλ$_{S}$dλ$_{E}$ D ( t )( Q ⊗ B ) . dξ$_{S}$dξ$_{E}$ = dt i $_{ℏ}$, dλ$_{S}$dλ$_{E}$ = − dt i $_{ℏ}$, (5)    dρ$_{S}$ = dt i $_{ℏ}$[ H$_{S}$, ρ$_{S}$ ] + dξ$_{S}$Qρ$_{S}$ + dλ$_{S}$ρ$_{S}$Q dρ$_{E}$ = dt i $_{ℏ}$[ H$_{E}$, ρ$_{E}$ ] + dξ$_{E}$Bρ$_{E}$ + dλ$_{E}$ρ$_{E}$B (3) Λ$_{stat}$ = Tr { ( D $^{†}$( t ) − D $^{†}$( t ) ) ( D ( t ) − D ( t ) ) } = Tr { D $^{†}$D ( t ) } − Tr { D ( t ) $^{2}$} . (6) A. Quantum Monte-Carlo formulation of Open Quantum Systems