Introduction

Actual quantum systems are open: they unavoidably interact, even if slightly, with their surrounding environment1. A useful phenomenological approach is to conceptualize the interaction as a generalized measurement performed by the environment onto the system2,3. As a consequence, the state vector of an open system follows stochastic trajectories in the Hilbert space of its stylized isolated counterpart. These trajectories are characterized by sudden transitions, quantum jumps. Since the experimental breakthroughs4,5 quantum jumps have been observed in atomic and solid-state single quantum systems under indirect measurement see e.g., ref. 1 for an overview. Quantum trajectory theory6,7,8,9,10,11,12 (see also refs. 2,3 for textbook presentations) connects these experimental results to the axiomatic theory of continuous measurement13,14 which derives deterministic master equations governing the dynamics of a quantum system open also in consequence of the perturbation due to measurement. According to quantum trajectory theory, the state operator of any open system whose evolution is governed by the Lindblad–Gorini–Kossakowski–Sudarshan equation2,3,15 can be “unraveled” i.e., represented as a statistical average over random realizations of post-measurement states. Mathematically, the average is computed over the realizations of a stochastic process describing the effect on the system of the interaction with an environment subject to continuous monitoring by a measurement device2,3,7,10,16. The precise definition of the stochastic process is contextual to the measurement scheme. Different measurement schemes result in distinct unravelings. In all cases, the stochastic process’ evolution law subsumes unitary dynamics with random collapses of the state vector occurring in consequence of an indirect measurement whilst continuously preserving the system’s Bloch hyper-sphere. Finally, in order to permit a measurement interpretation, the stochastic process must be non-anticipating: the statistics up to the present observation must be invariant with respect to future measurement events17,18. The theory of quantum trajectories is still under active development, see e.g., refs. 19,20. Recent experiments even support the possibility of using the theory to identify precursors of the imminent occurrence of a jump21.

Time-local and trace-preserving master equations encountered in applications belong to a class larger than that specified by the Lindblad–Gorini–Kossakowski–Sudarshan form. This more general class consists of master equations generating completely bounded maps. It is well known (see e.g., ref. 22) that the fundamental solution of a Lindblad–Gorini–Kossakowski–Sudarshan master equation is completely positive: it maps any positive operator on the system’s Hilbert space, and eventually any extension of it by the tensor product with the identity map on an auxiliary arbitrarily sized Hilbert space, into a positive operator. Completely bounded maps are those in the larger class defined by weakening the positivity requirement into that of boundedness23.

Master equations with completely bounded fundamental solution are typically obtained by time convolutionless perturbation theory24 or by tracing environment degrees of freedom in Gaussian system-environment models of Bosons e.g., refs. 25,26 or Fermions e.g., refs. 27,28, and in other exactly integrable models of system-environment interactions e.g., ref. 29. We refer the reader to30 for a discussion of the reasons why master equations generating completely bounded maps are considered in applications, including a phenomenological discussion of the domain of validity. More theoretical arguments upholding the physical relevance of completely bounded maps directly from the postulates of quantum mechanics are presented in refs. 31,32,33,34,35 see, however, also ref. 36.

The importance of extending quantum trajectory theory to master equations generating completely bounded maps has been recognized early in the literature see e.g., § 9 of ref. 2. Existing frameworks for unraveling, however, require either the introduction of ancillary Hilbert spaces37 or to postulate memory and prescience effects between trajectories38,39 without a measurement interpretation40,41.

Here, we prove that master equations generating completely bounded maps admit an unraveling in non-anticipating quantum trajectories on the system’s Bloch hyper-sphere as it is the case under the more restrictive hypothesis of complete positivity. Specifically, we show that the quantum trajectories are realizations of the solutions of a system of ordinary stochastic differential equations driven by counting processes. The only requirement is that each realization of a quantum trajectory enters the Monte-Carlo average with its own weight factor. The weight factor is a martingale μt, a stochastic process whose expectation value is conserved on average (see e.g., ref. 42) so to ensure trace preservation. At any time t, the martingale μt obeys on its turn an ordinary stochastic differential equation enslaved to those governing the state vector of the system. For reasons that will become clear, we refer to μt as the “influence martingale”.

We illustrate our main result in integrable models whose unravelings in the Hilbert space of the system was previously believed to hinge upon memory and prescience effects or simply not possible because of non-positive preserving dynamics (Redfield equation). Besides a measurement interpretation, unravelings provide a numerical avenue to integrate open quantum systems in high dimensional Hilbert spaces9. In particular, it is well known (see e.g., chapter 7 of ref. 2) that the integration times of the state operator respectively computed from the master equation and from an average over quantum trajectories is expected to undergo a cross-over as the dimension of the Hilbert space increases. We verify the existence of the cross-over in a test case using QuTiP43, a widely applied toolbox for efficient numerical simulations of open quantum systems. We emphasize that the convergence of unravelings based on the influence martingale is guaranteed by the well-established theory of ordinary stochastic differential equations with jumps see e.g., ref. 44.

Finally, we show how the influence martingale naturally accounts for photo-current oscillations which are observed in experimental quantum optics.

Results

Given a microscopic unitary dynamics, a partial trace implemented for example with the help of time convolutionless perturbation theory24 (see also e.g., chapters 9 and 10 of ref. 2) yields

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}=-\imath \,\left[{{{{{{{{\rm{H}}}}}}}}}_{t},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}\right]+\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}\frac{\left[{{{{{{{{\rm{L}}}}}}}}}_{\ell },{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }\right]+\left[{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t},{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }\right]}{2}$$
(1)

The master equation (1) embodies the universal form of a time-local and trace-preserving evolution law.

In (1), the Hamiltonian Ht is the generator of a unitary dynamics. The collection \({\{{{{{{{{{\rm{L}}}}}}}}}_{\ell }\}}_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}\) consists of so-called Lindblad operators modeling the interaction with the environment. The weights Γℓ,t’s are related to the probability per unit of time of the collapse associated to the Lindblad operators they couple to (1). Khalfin’s theorem45 forbids exponential decay in quantum mechanics outside the intermediate asymptotic singled out by the weak coupling scaling limit14,46. Thus the Γℓ,t’s are in general time dependent functions with arbitrary sign. We only require them to be bounded. The celebrated Lindblad-Gorini-Kossakowski-Sudarshan master equation is thus a special case of (1) corresponding to the complete positivity conditions

$${{{\Gamma }}}_{\ell ,t}\,\ge \,0\,\,\,\,\,\,\ell =1,\ldots ,{{{{{{{\mathscr{L}}}}}}}}.$$
(2)

The conditions (2) are usually derived from microscopic models in the weak coupling limit14. We emphasize that even when the conditions (2) do not hold, (1) may still admit completely positive solutions but only for special initial states and initial times. These cases correspond to completely positive but not completely positive divisible dynamical maps22,47.

The gist of the proof of the unraveling of (1) via the influence martingale is based on an extension of Girsanov’s change of measure formula, a well known result in the theory of stochastic processes (see e.g., ref. 42). Roughly speaking, Girsanov formula expresses the average of a generic functional Ft up to time t of a stochastic process in terms of the weighted average of the same functional now evaluated over a second distinct stochastic process

$$\tilde{{{{{{{{\rm{E}}}}}}}}}({F}_{t})={{{{{{{\rm{E}}}}}}}}({M}_{t}{F}_{t}).$$
(3)

Here E and \(\tilde{{{{{{{{\rm{E}}}}}}}}}\) denote the expectation values with respect to the probability measures of the two stochastic processes. Girsanov’s theorem states that the scalar weighing factor Mt must be a positive definite martingale42 satisfying for all t

$${{{{{{{\rm{E}}}}}}}}({M}_{t})=1.$$
(4)

The extension we propose consists in relinquishing the requirement that the martingale be positive definite. We will return below on the interpretation of the negative values of the influence martingale. We now turn to detail the proof of the unraveling.

Our aim is to prove that a state operator solution of (1) always admits the representation

$${{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}={{{{{{{\rm{E}}}}}}}}\left({\mu }_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right).$$
(5)

Here E denotes the expectation value operation, ψt is a stochastic state vector at time t defined in the Hilbert space of the system and \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\) is its adjoint dual. Finally, μt is a scalar stochastic process enjoying the martingale property. We prescribe ψt, \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\) and μt to obey evolution laws such that the expectation value (5) indeed solves (1).

First, we require that the state vector solve the Itô stochastic differential equation9

$${{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}= -\imath \,{{{{{{{{\rm{H}}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{\rm{d}}}}}}}}t-\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }-{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{\rm{d}}}}}}}}t\\ +\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}}{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }-{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right)$$
(6)

In (6), the \({\{{\nu }_{\ell ,t}\}}_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}\) are a collection of counting processes (see e.g., refs. 2,3,7,16). The statistics of the counting process increments dνℓ,t’s are fully specified for \(\ell ,{{{{{{{\mathscr{k}}}}}}}}=1,\ldots ,{{{{{{{\mathscr{L}}}}}}}}\) by

$${{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}{{{{{{{\rm{d}}}}}}}}{\nu }_{{{{{{{{\mathscr{k}}}}}}}},t}={\delta }_{\ell ,{{{{{{{\mathscr{k}}}}}}}}}{{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}$$
(7)
$${{{{{{{\rm{E}}}}}}}}\left(\right.{{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}\left|\right.{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\left)\right.={{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{\rm{d}}}}}}}}t$$
(8)

where \({\{{{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}\}}_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}\) is a collection of strictly positive definite functions of time. Equation (7) states that dνℓ,t can only take values 0 or 1. The conditional expectation \({{{{{{{\rm{E}}}}}}}}({{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}| {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })\) is called the compensator of the counting process νℓ,t and determines the jump rate given the values of the state vector and its complex adjoint at time t42. The equations governing \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\) follow immediately from (6) by applying the complex adjoint operation. We associate to (6) and to the equation for the adjoint, initial data on the Bloch hyper-sphere i.e., \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}=\parallel {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}{\parallel }^{2}=1\). We emphasize that the stochastic Schrödinger equation (6) and the counting processes (7), (8) are essentially the same as in ref. 9.

Next, we need μt to obey an evolution law admitting solutions enjoying the martingale property. We thus require μt to evolve according to the Itô stochastic differential equation

$${{{{{{{\rm{d}}}}}}}}{\mu }_{t}={\mu }_{t}\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\left(\frac{{{{\Gamma }}}_{\ell ,t}}{{{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}}-1\right){{{{{{{{\rm{d}}}}}}}}\iota }_{\ell ,t}$$
(9)
$${{{{{{{{\rm{d}}}}}}}}\iota }_{\ell ,t}={{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}-{{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{\rm{d}}}}}}}}t$$
(10)
$${\mu }_{0}=1$$
(11)

The solution of an equation of the form (9) is by construction a local martingale (see e.g., ref. 42 for details). Namely, the source of randomness in (9) are the innovation processes (10)7,16 defined by compensating counting process increments dνℓ,t’s by their conditional expectation (8). The immediate consequence is that the expectation value of the increments of μt conditional on the values of ψt, \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\) vanishes at any time instant t:

$${{{{{{{\rm{E}}}}}}}}({{{{{{{\rm{d}}}}}}}}{\mu }_{t}| {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })=0.$$
(12)

A local martingale becomes a strict martingale, i.e., satisfies the condition Eμt = 1 for all t, if the integrability condition \({{{{{{{\rm{E}}}}}}}}\mathop{\sup }\limits_{t}| {\mu }_{t}| \; < \; \infty\) holds. In practice, we expect μt to be a strict martingale if all Γℓ,t’s are bounded functions of t during the evolution horizon. We take for granted this physically reasonable condition, and therefore that the process μt is a strict martingale.

The last step in the proof of the unraveling via the influence martingale is to compute the differential of the expectation value (5) using (6), (7), (8), and (9). A straightforward application of stochastic calculus proves that the state operator satisfies the time local master Eq. (1). We report the details of the calculation in Methods.

The question naturally arises whether the evolution law (6) preserves the squared norm of the stochastic process ψt, and, as a consequence, justifies the interpretation of ψt as state vector of the system. We verify that the squared norm satisfies the Itô stochastic differential equation

$${{{{{{{\rm{d}}}}}}}}\left({\left\Vert {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}\right)=\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\left({{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}-{{{\Gamma }}}_{\ell ,t}{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{\rm{d}}}}}}}}t\right)\left(1-{\left\Vert {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}\right).$$
(13)

Therefore, for arbitrary initial values ψ0, \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}^{{{{\dagger}}} }\) the expected value of the squared norm is not preserved unless (2) holds true. Nevertheless, the Bloch hyper-sphere (i.e., the manifold ∥ψt∥2 = 1) is preserved by the dynamics. Thus, we can interpret ψt as a state vector for any quantum trajectory evolving from physically relevant initial data assigned on the Bloch hyper-sphere. A further useful consequence is that Bloch hyper-sphere valued solutions of (6) can always be couched into the form of the ratio of the solution of a linear stochastic differential equation divided by its norm. We defer the proof of the claim to Methods.

Some observations are in order regarding equations (6) and (9). The evolution law (9) of the influence martingale is enslaved to that of the state vector (6): μt exerts no feedback on the stochastic Schrödinger equation (6). Most importantly, the state vector is a Markov process and the influence martingale is also non anticipating. This is intuitively pleasing because in any finite dimensional Hilbert space the master equation (1) is just a matrix-valued time non-autonomous linear ordinary differential equation. Finally, we emphasize the different nature of the weights Γℓ,t’s, and of the positive definite rates \({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}\)’s. The former ones are theoretical predictions fixed by the microscopic dynamics. The \({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}\)’s are either inferred from experimental measurement or, in numerical applications, selected based on integration convenience. Such arbitrariness reflects the fact that quantum trajectories generated by an unraveling exist only contextually to a setup or in the language of48 are “subjectively real”.

Let us now turn to the interpretation of the influence martingale. Girsanov’s formula (see e.g., ref. 42) states that, when the martingale process μt in (5) is positive definite, it specifies a change of probability measure. The influence martingale can, however, take negative values when the Γℓ,t’s do so. As μt is non-anticipating, at any time t it is always possible to represent it as the difference of two positive definite and correlated processes

$${\mu }_{t}^{(\pm )}=\max (0,\pm {\mu }_{t}).$$
(14)

The immediate consequence is that we can couch (5) into the form

$${{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}={{{{{{{\rm{E}}}}}}}}\left({\mu }_{t}^{(+)}{{{\boldsymbol{{\psi }}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }-{\mu }_{t}^{(-)}{{{\boldsymbol{{\psi }}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right)$$
(15)

Using the explicit expression of the influence martingale and of the state vector in terms of the fundamental solution of the linear dynamics (“Methods”) it is straightforward to verify that the argument of the expectation value is the difference of two completely positive dynamical maps. We thus recognize that from the mathematical point of view, the need to introduce the influence martingale naturally stems from the general result in linear operator algebra known as the Wittstock–Paulsen decomposition23 stating that any completely bounded map is always amenable to the difference of two completely positive maps.

If one insists on the change of probability measure interpretation, negative values of the influence martingale would imply that some realizations of quantum trajectories in the mathematical path-space should be weighed by a “negative probability”. Exactly for the reasons put forward by Feynman in ref. 49, even such interpretation does not pose any logical difficulty when the initial state operator belongs to the compatibility domain of operators whose positivity is preserved by the evolution33,34. Negative values of the influence martingale only contribute as an intermediate step to the Monte Carlo evaluation of the state operator. In other words, they do not specify “the final probability of verifiable physical events”49. We refer to refs. 50,51 for a mathematically rigorous operational definition of negative probabilities in quantum mechanics recently developed starting from Feynman’s argument.

From a more qualitative point of view, it is suggestive to interpret negative values of μt as a form of interference that occurs in the mathematical path-space in order to ensure the convergence of a Monte Carlo average to a completely bounded deterministic dynamical map. In ref. 49 (page 246–48) Feynman shows how interference patterns in a double slit experiment can be formally computed by means of arithmetic averages also including events weighed by a negative probability. The Lindblad weights Γℓ,t’s in the master equation are essentially time derivatives of survival probabilities. These latter quantities may take negative values in consequence of a phenomenon usually interpreted as re-scattering of decay products from the environment to the system46. Putting together these admittedly heuristic considerations, motivates the appeal of conceptualizing the influence martingale as the expression of a form of environment feedback-induced interference of quantum trajectories. By the same token, we motivate its name with a role reminiscent of the influence functional introduced by Feynman and Vernon in ref. 25. To dispel any possible misunderstanding, however, we wish to add that the above considerations do not aim at arguing a one-to-one correspondence between the phenomenon of quantum interference and completely bounded maps.

Examples

In low dimensional Hilbert spaces, algorithms based on quantum trajectories are not expected to bring any numerical efficiency advantage with respect to direct integration of the master equation. Thus the purpose of the examples is only to highlight how a non-anticipating unraveling reproduces physical phenomena such as quantum revivals usually attributed to memory and prescience effects. In all the examples we set for convenience \({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}=| {{{\Gamma }}}_{\ell ,t}|\) in (8).

The master equation in Dirac’s interaction picture of a two level atom in a photonic band gap29,52 is

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}=\frac{{S}_{t}}{2\,\imath }\left[{\sigma }_{+}{\sigma }_{-},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}\right]+{{{\Gamma }}}_{t}\frac{\left[{\sigma }_{-}{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t},{\sigma }_{+}\right]+\left[{\sigma }_{-},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{\sigma }_{+}\right]}{2}$$
(16)

where σ± = (σ1 ± ı σ2)/2 and \({\left\{{\sigma }_{i}\right\}}_{i = 1}^{3}\) are Pauli matrices. The time dependent functions St and Γt are respectively the Lamb shift and the Lindblad weight factor. Negative values of Γt also imply a violation of the Kossakowski conditions (see e.g., refs. 22,47) a weaker form of positivity that might be imposed on ρt53. This fact renders the unraveling of (16) in quantum trajectories particularly probing.

In order to explore a genuine strong system-environment coupling, we proceed as in38. Let g and e be respectively the ground and excited state of σ+σ−. Using the solution of the off-diagonal matrix element e†ρtg = cte†ρ0g of a qubit in a photonic band gap (equation (2.21) of ref. 29 with β = −δ), we determine the Lamb shift St and weight factor Γt by2

$${S}_{t}=-2\,{{{{{{{\rm{Im}}}}}}}}\frac{{\dot{c}}_{t}}{{c}_{t}},\,\,\,{{{\Gamma }}}_{t}=-2\,{{{{{{{\rm{Re}}}}}}}}\frac{{\dot{c}}_{t}}{{c}_{t}}$$
(17)

We also translate the origin of time to t ≈ 1.4 so that St and Γt vanish at time origin.

In Fig. 1a, we show the time dependence of the Lindblad weight factor Γt and the Lamb shift St. In Fig. 1c, d, we show typical realizations of μt. In particular, Fig. 1d exhibits the exponential growth of μt when Γt is negative and in the absence of jumps. On the other hand, when Γt is positive μt is constant in between jumps. Finally, in Fig. 1c we show a realization of μt taking negative values in consequence of a quantum jump.

Fig. 1: Illustration of qubit state and influence martingale trajectories.
figure 1

a Time dependence of the weight function Γt (full line) and the Lamb shift St (dashed), defined as in (17). b–d Different realizations of the stochastic evolution displaying \({\left\Vert {{{{{{{{\boldsymbol{e}}}}}}}}}^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}\) (diamonds), \({\mu }_{t}{\left\Vert {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}\) (crosses), and μt (full line), where the system state ψt and influence martingale according to μt (6) and (9). The initial data are \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}=({{{{{{{\boldsymbol{e}}}}}}}}+{{{{{{{\boldsymbol{g}}}}}}}})/\sqrt{2}\) where g and e are respectively the ground and excited states of H = σ+ σ−.

Full size image

Figure 2 reports the result of our numerical integration for distinct values of the initial data. We generate the quantum trajectories by mapping (6) into a linear equation as described in Methods. The (black) full lines always denote predictions from the master equation. Monte Carlo averages are over ensembles of 104 realizations. We theoretically estimate errors with twice the square root of the ensemble variance of the indicator of interest. In all cases, Monte Carlo averages and master equation predictions are well within fluctuation-induced errors. The occurrence of “quantum revivals” can be also quantitatively substantiated observing that the measure of system-environment information flow introduced in ref. 54 is simply related to Γt for this model47,54. Namely, equation (66) of ref. 47 relates the direction of the information flow in the model to the sign of the time derivative of ∣Γt∣. We perform the numerical integration using the Tsitouras 5/4 Runge-Kutta method automatically switching for stiffness detection to a 4th order A-stable Rosenbrock. The Julia code is offered ready for use in the “DifferentialEquation.jl” open source suite55.

Fig. 2: Illustration of the influence martingale for a qubit in a photonic band gap.
figure 2

Monte Carlo averages versus master equation predictions for (16). The scatter plots show \({{{{{{{{\boldsymbol{f}}}}}}}}}_{1}^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{{\boldsymbol{f}}}}}}}}}_{1}\), with \({{{{{{{{\boldsymbol{f}}}}}}}}}_{1}=({{{{{{{\boldsymbol{e}}}}}}}}+{{{{{{{\boldsymbol{g}}}}}}}})/\sqrt{2}\), with initial states ψ0 = f1 (diamonds) and \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}={{{{{{{{\boldsymbol{f}}}}}}}}}_{2}=({{{{{{{\boldsymbol{e}}}}}}}}-{{{{{{{\boldsymbol{g}}}}}}}})/\sqrt{2}\) (x's) and their sum (crosses), where e and g are the excited and ground state of σ+σ−, respectively. The data is obtained from 104 realizations. The continuous lines show the solutions obtained from directly integrating the master equation. The shaded area is determined by two times the square root of the ensemble variance of the indicator of interest. The starting time mesh for the adaptive code is dt = 0.03.

Full size image

Next, we study a qubit model with “controllable positivity”, whose dynamics are governed by the master equation

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}=\mathop{\sum }\limits_{\ell =1}^{3}{{{\Gamma }}}_{\ell ,t}({\sigma }_{\ell }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{\sigma }_{\ell }-{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}).$$
(18)

The above equation provides a mathematical model of an all-optical setup exhibiting controllable transitions from positive-divisible to non-positive divisible evolution laws56. We focus on the case when the Lindblad weights are of the form (ℓ = 1, 2, 3)

$${{{\Gamma }}}_{\ell ,t}=-{a}_{\ell }+{b}_{\ell }\tanh ({c}_{\ell }\,t)$$
(19)

We choose aℓ, bℓ, cℓ > 0 such that all Lindblad weights are negative during a finite time interval around t = 0. As a consequence, not only the Kossakowski conditions are violated but the rate operator57 has negative eigenvalues for the initial conditions we consider. Under these hypotheses, and to the best of our knowledge, only the influence martingale permits to unravel the master equation in quantum trajectories taking values in the Hilbert space of the system. Figure 3 shows the evolution in time of e†ρte where e is the excited state of the Pauli matrix σ3, i.e., it has eigenvalue 1. The full (black) line is the master equation prediction, the crosses show the Monte Carlo average (5) for 103 realizations and the diamonds for 104. The light green shaded area shows twice the square root of the variance for 103 realizations and the darker brown for 104. In agreement with the theory, increasing the number of realizations in the ensemble evinces convergence.

Fig. 3: Illustration of the influence martingale for the random unitary model.
figure 3

Monte Carlo averages versus master equation predictions for the energy level populations of the random unitary model (18). The Lindblad weights are \({{{\Gamma }}}_{1,t}=-0.5+2\tanh (\sqrt{2}\,t)\), \({{{\Gamma }}}_{2,t}=-1+2\tanh (\sqrt{3}\,t)\) and \({{{\Gamma }}}_{3,t}=-0.8+2\tanh (\sqrt{5}\,t)\). The initial condition of the qubit is \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}=\frac{\sqrt{3}}{2}{{{{{{{\boldsymbol{e}}}}}}}}+\frac{1}{2}{{{{{{{\boldsymbol{g}}}}}}}}\) where g, e are the ground and excited state of σ3, respectively. The black full line gives the prediction by solving the master equation (18). The crosses show the stochastic average after 103 realizations and the (light) green shaded area displays fluctuation-related errors estimated as in Fig. 2. Similarly, the diamonds show the ensemble average after 104 realizations and the (dark) brown shaded area the estimated error.

Full size image

As a further example of how the influence martingale associates a quantum trajectory picture even to master equations with non positive definite solutions, we consider a Redfield equation model. We call Redfield a master equation of the form (1) obtained from an exact system-environment dynamics by implementing the Born–Markov approximation without a rotating wave approximation. Redfield equations are phenomenologically known to give accurate descriptions of the system dynamics at arbitrary system-environment coupling although they do not guarantee a positive time evolution of the state operator see e.g., refs. 2,30,58. In “Methods”, we outline the derivation of a Redfield equation from the exact dynamics of two non-interacting qubits in contact with a boson environment59. The result is an equation of the form (1) with only two Lindblad operators \({\left\{{{{{{{{{\rm{L}}}}}}}}}_{\ell }\right\}}_{\ell = 1}^{2}\) satisfying the commutation relations

$$[{{{{{{{{\rm{L}}}}}}}}}_{\ell },{{{{{{{{\rm{L}}}}}}}}}_{{{{{{{{\mathscr{k}}}}}}}}}^{{{{\dagger}}} }]={\delta }_{\ell {{{{{{{\mathscr{k}}}}}}}}},\,[{{{{{{{{\rm{L}}}}}}}}}_{\ell },{{{{{{{{\rm{L}}}}}}}}}_{{{{{{{{\mathscr{k}}}}}}}}}]=0.$$

The corresponding weights are time independent

$${{{\Gamma }}}_{\ell ,t}={\lambda }_{\ell }=\frac{{\gamma }_{1}+{\gamma }_{2}+{(-1)}^{\ell }\sqrt{2}\sqrt{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+2\,{\kappa }^{2}}}{4}$$
(20)

where γ1, γ2, κ are real numbers. We refer to Methods for the explicit expressions of H and \({\left\{{{{{{{{{\rm{L}}}}}}}}}_{\ell }\right\}}_{\ell = 1}^{2}\). Inspection of (20) shows that λ1 is negative definite. Figure 4 shows how the influence martingale reproduces the predictions of the Redfield equation.

Fig. 4: Illustration of the influence martingale for a Redfield equation.
figure 4

The parameters for the model (20) are γ1 = 1, γ2 = 4, α = 3 and κ = 1. The initial condition is \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{0}=\sqrt{0.2}\,{{{{{{{{\boldsymbol{w}}}}}}}}}_{1}+\sqrt{0.1}\,{{{{{{{{\boldsymbol{w}}}}}}}}}_{2}+\sqrt{0.7}\,{{{{{{{\boldsymbol{g}}}}}}}}\) where g is the ground state, \({{{{{{{{\boldsymbol{w}}}}}}}}}_{1}={L}_{1}^{{{{\dagger}}} }\,{{{{{{{\boldsymbol{g}}}}}}}}\) and \({{{{{{{{\boldsymbol{w}}}}}}}}}_{2}={L}_{2}^{{{{\dagger}}} }\,{{{{{{{\boldsymbol{g}}}}}}}}\). The crosses, diamonds, and x's show the result from Monte Carlo simulations after 104 realizations. The full lines are master equation predictions and the shaded regions have the same meaning as in Fig. 2. The time mesh is dt = 0.0125. The diamonds show that ρt is non positive definite for t > 3.

Full size image

Numerical application

Simulating Large Quantum Systems was the motivation of the authors of ref. 9 for unraveling a master equation or, in their words, for “Monte Carlo wave-function methods”. The reason is the following. Given a \({{{{{{{\mathscr{N}}}}}}}}\)-state system numerical integration of the master equation requires to store \(O({{{{{{{{\mathscr{N}}}}}}}}}^{2})\) real numbers so that the computing time typically scales as \(O({{{{{{{{\mathscr{N}}}}}}}}}^{4})\). Unraveling the state operator as the average over state vectors generated by a Markov process requires to store only \(O(2\,{{{{{{{\mathscr{N}}}}}}}})\) real numbers for each realization. This means that the computing time scales as \(O({{{{{{{{\mathscr{N}}}}}}}}}^{2}\,\times \,{{{{{{{\mathscr{M}}}}}}}})\) where \({{{{{{{\mathscr{M}}}}}}}}\) is the number of realizations or just \(O({{{{{{{{\mathscr{N}}}}}}}}}^{2})\) on a parallel processor3. Thus, in high dimensional Hilbert spaces, also due to the existence of efficient numerical algorithms for stochastic differential equations44, ensemble averages are expected to offer a real advantage for numerical computation2,3,48. It is worth noticing that the reasoning is very similar to that motivating the use of Lagrangian in place of Eulerian numerical methods in the context of classical hydrodynamics, see e.g., ref. 60.

To exhibit that the argument applies to the influence martingale, we compare computing times versus the dimension of the Hilbert space using QuTiP43, a standard numerical toolbox for open quantum systems. Specifically, we take a chain of \({{{{{{{\mathscr{N}}}}}}}}\) coupled qubits and directly integrate the master equation using QuTiP. Next, we perform the same calculation using the QuTiP package for Monte Carlo wave-function methods combined with the implementation of the influence martingale. To do so we choose the rates in (8) as we detail below.

In (1) we take \({{{{{{{\mathscr{L}}}}}}}}=2\,{{{{{{{\mathscr{N}}}}}}}}\) where the Lindblad operator Lℓ for \(\ell =1,\ldots ,{{{{{{{\mathscr{N}}}}}}}}\) is the tensor product of the lowering operator of the ℓth qubit with the identity acting on the Hilbert space of the remaining qubits. Thus we set

$${{{{{{{{\rm{L}}}}}}}}}_{\ell }={\sigma }_{-}^{(\ell )},\,\,\,{{{{{{{{\rm{L}}}}}}}}}_{\ell +{{{{{{{\mathscr{N}}}}}}}}}={\sigma }_{+}^{(\ell )}\,\,\,\ell =1,\ldots ,{{{{{{{\mathscr{N}}}}}}}}$$
(21)

The Hamiltonian is

$${{{{{{{\rm{H}}}}}}}}=\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{N}}}}}}}}}{\sigma }_{+}^{(\ell )}{\sigma }_{-}^{(\ell )}+\lambda \mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{N}}}}}}}}-1}({\sigma }_{+}^{(\ell )}{\sigma }_{-}^{(\ell +1)}+{\sigma }_{+}^{(\ell +1)}{\sigma }_{-}^{(\ell )})$$
(22)

For the sake of simplicity, we assume that the Lindblad weights in (1) are all strictly positive definite with the only exception of Γ1,t which can take negative values. For any ℓ different from 1 and \({{{{{{{\mathscr{N}}}}}}}}+1\) we choose the rates of the counting processes (8)

$${{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}={{{\Gamma }}}_{\ell ,t} > 0\,\,\,\,\,\ell \ne 1,{{{{{{{\mathscr{N}}}}}}}}+1$$
(23)

Next, we use the fact that given a collection of scalars \({{{{{{{{\mathscr{w}}}}}}}}}_{\ell }\) on the real axis, it is always possible to find an equal number of positive definite scalars \({{{{{{{{\mathscr{r}}}}}}}}}_{\ell }\) and a negative definite real c such that the set of equations

$${{{{{{{{\mathscr{w}}}}}}}}}_{\ell }={{{{{{{{\mathscr{r}}}}}}}}}_{\ell }+c$$
(24)

is satisfied. We, therefore, set for any t

$${{{\Gamma }}}_{1,t}={{{{{{{{\mathscr{r}}}}}}}}}_{1,t}+{c}_{t}$$
(25)
$${{{\Gamma }}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}={{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}+{c}_{t}$$
(26)

where now \({{{{{{{{\mathscr{r}}}}}}}}}_{1,t}\) and \({{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}\) specify the rates of the counting processes dν1,t and \({{{{{{{\rm{d}}}}}}}}{\nu }_{1+{{{{{{{\mathscr{N}}}}}}}},t}\). As the Lindblad operators satisfy

$${\sigma }_{+}^{(1)}{\sigma }_{-}^{(1)}+{\sigma }_{-}^{(1)}{\sigma }_{+}^{(1)}={{{{{{{{\rm{1}}}}}}}}}_{{{{{{{{\mathcal{H}}}}}}}}}$$
(27)

on the Bloch hyper-sphere (∥ψt∥2 = 1) we simplify the drift in (6) using the identity

$${{{\Gamma }}}_{1,t}\frac{{\sigma }_{+}^{(1)}{\sigma }_{-}^{(1)}-{\left\Vert {\sigma }_{-}^{(1)}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}+{{{\Gamma }}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}\frac{{\sigma }_{-}^{(1)}{\sigma }_{+}^{(1)}-{\left\Vert {\sigma }_{+}^{(1)}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\\ ={{{{{{{{\mathscr{r}}}}}}}}}_{1,t}\frac{{\sigma }_{+}^{(1)}{\sigma }_{-}^{(1)}-{\left\Vert {\sigma }_{-}^{(1)}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}+{{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}\frac{{\sigma }_{-}^{(1)}{\sigma }_{+}^{(1)}-{\left\Vert {\sigma }_{+}^{(1)}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}$$
(28)

We conclude that the state operator evolves on the Bloch hyper-sphere according to the same stochastic Schrödinger equation of ref. 9. We can therefore directly use the Monte Carlo wave function package of QuTiP for computing the evolution of the state vector. Using this information, we compute the influence martingale for each trajectory and finally the state operator. We refer to Methods for further details.

Figure 5 shows the populations of several sites for \({{{{{{{\mathscr{N}}}}}}}}=11\). The computation time is shown in Fig. 6a. At \({{{{{{{\mathscr{N}}}}}}}}=10\) we observe a cross-over of the computation time curves. After \({{{{{{{\mathscr{N}}}}}}}}=11\) the influence martingale based algorithm becomes more efficient without applying any adapted optimization. From 12 qubits on, direct integration of the master equation becomes unwieldy (Apple M1 CPU). With the influence martingale, even 13 coupled qubits take only a few minutes of computation time. Most of the computation time is due to generating the trajectories. The actual computation of the martingale takes 0.19s for \({{{{{{{\mathscr{N}}}}}}}}=2\) and 0.42s for \({{{{{{{\mathscr{N}}}}}}}}=11\). The number of realizations is always 103. Figure 6b shows the root mean square error of site occupations averaged over all sites. The error stays approximately constant when \({{{{{{{\mathscr{N}}}}}}}}\) increases.

Fig. 5: Site populations in a long qubit chain.
figure 5

Population of sites 1, 4, and 11 for the qubit chain (22) with L = 11. The marks show the result of the stochastic evolution and the full black lines the result of numerically integrating the master equation. The weights are Γℓ,t = γ, Γℓ+N,t = δ for ℓ = 1, … N, Γ1+N,t = δ for all t, \({{{\Gamma }}}_{1,t}={{{\Gamma }}}_{1,t}=\gamma -12\exp (-2{t}^{3}){\sin }^{2}(15t)\) with γ = (1/0.129)(1 + 0.063) and δ = (1/0.129)0.063 and λ = 10.

Full size image
Fig. 6: Numerical performance of the influence martingale.
figure 6

a Computation time for both the Master Equation and Influence Martingale method as a function of the amount of qubits in the chain. For the stochastic method, we generated 1000 realizations. For \({{{{{{{\mathscr{N}}}}}}}} \; > \; 11\) the use of the master equation becomes unwieldy on our laptops whereas we were able to take further points using the unraveling. b The root mean square error of the populations averaged over all individual sites.

Full size image

Theoretical application

In experimental quantum optics, photo-current is usually defined as the average number of detection events per unit of time. The stochastic differentials (7) mathematically describe increments of the photocurrent associated to detection events corresponding to transitions induced by the Lindblad operator Lℓ in (1)3.

The study of time-dependent transport properties of photo-excited undoped super-lattices highlighted the phenomenon of photo-current oscillations61. Photo-current oscillations naturally come about when working with the time convolutionless master equation with non positive Lindblad weights. In such a case and under standard hypotheses (see “Methods”) it is straightforward to verify that a system governed by an Hamiltonian \({{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{0}}}\) when isolated satisfies when the coupled to an environment the energy balance equation

$${{{{{{{\rm{d}}}}}}}}\,{{{{{{{\rm{Tr}}}}}}}}({{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{o}}}{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t})=\frac{{{{{{{{\rm{Tr}}}}}}}}\left[{{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{o}}},{{{{{{{{\rm{H}}}}}}}}}_{t}\right]{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{\rm{d}}}}}}}}t}{\imath }+\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{\epsilon }_{\ell }\,{{{{{{{\rm{d}}}}}}}}({{{{{{{\rm{E}}}}}}}}{\mu }_{t}{\nu }_{\ell ,t})$$
(29)

where the ϵℓ are the energy quanta exchanged in transitions. The expression generalizes the results of ref. 3 showing that, as the strength of the system environment coupling increases, the average values of photo-current increments are modulated by the influence martingale in thermodynamic relations. The conclusion is that photo-current oscillations reflect a heat flow both from and to the system.

Discussion

In this paper, we prove the existence of a non-anticipating unraveling of time local master equations whose fundamental solution is a completely bounded map. General results in operator algebra show that completely positive maps are a particular case of completely bounded ones. Accordingly, the unraveling we present naturally recovers the well known theory when the hypothesis of complete positivity is added. Specifically, the stochastic state vector obeys a Markovian evolution on the Bloch hyper-sphere of the system. The distinctive feature which makes the unraveling possible consists in realizing that outer products of the stochastic state vector must be weighed by a non-positive definite scalar martingale in order to generate Monte Carlo averages converging to a completely bounded map. This is a major conceptual difference with the martingale methods previously considered under the hypothesis of a completely positive divisible dynamics (see e.g., refs. 7,16,62). There the martingale is strictly positive because it is the norm squared of a complex vector. Furthermore, the introduction of a martingale weight factor serves the purpose of couching norm preserving state vectors obeying a non-linear evolution in terms of vectors satisfying linear stochastic differential equations.

In ref. 48 Wiseman discussed three interpretations of quantum trajectories generated by unravelings. Here, exactly as in48, the order of listing is absolutely not meant to reflect importance. The first interpretation is as mathematical tool to compute the solution of the Lindblad–Gorini–Kossakowski–Sudarshan equation for high dimensional systems (see e.g., ref. 9). The second interpretation of quantum trajectories construes them as subjectively real: their existence and features are determined only contextually to a given physical setup (see e.g., refs. 12,63). And finally, quantum trajectories might be an element of a still missing theory of quantum state reduction15,64,65,66. We believe that the influence martingale yields a significant contribution to, at least, the first two interpretations. As a mathematical tool, the unraveling via influence martingale method only involves the use of ordinary stochastic differential equations driven by counting processes. Counting processes are paradigmatic mathematical models of measurement-induced state vector collapse as they naturally describe individual detection events while permitting straightforward numerical implementation2,3,7,16. The proof of the unraveling is rigorous and straightforward. The convergence of Monte Carlo averages is guaranteed by standard results in the theory of stochastic differential equations44. A further advantage is that the unraveling does not rely on any hypothesis on the sign of the scalar prefactors, weights, of the Lindblad operators in the master equation. Generalizing what was previously established for master equations derived in the weak coupling scaling limit2,3, we here provide explicit evidence of the advantage of using the influence martingale to integrate a substantially larger class of master equations when the dimension of the Hilbert space is large. Finally, we observe that if the purpose of introducing quantum trajectories is limited to numerical applications, resorting to the generation of ostensible statistics as in ref. 67 may further speed up calculations. For ostensible statistics trace preservation in (5) holds not pathwise but only on average, Bloch hyper-sphere conservation is thus not required and the influence martingale can be replaced by a simple jump process.

Regarding the second interpretation, the meaning of the influence martingale is that of representing the completely bounded fundamental solution of the universal form of the time local master equation as an average over stochastic realizations of completely bounded maps. General results23 in linear operator algebra prove that a completely bounded map can be embedded into a completely positive map acting on a larger Hilbert space. Combining this fact with the non-anticipating nature of the unraveling guarantees the measurement interpretation (“Methods”). An explicit example of the construction of the embedding is given in ref. 37. There the unraveling was only defined in the extended Hilbert space thus requiring the introduction of a “minimal" dynamics for an auxiliary environment. Here we prove that quantum trajectories can be computed directly on the Bloch hyper-sphere of the system, their occurrence being always consistent with a measurement performed on an environment that does not need to be specified.

In conclusion, the main result of the present paper substantially extends the domain of application of quantum trajectory-based methods of state and dynamical parameter estimation, prediction and retrodiction as recently reviewed in ref. 68.

Methods

Derivation of the master equation

In order to verify that the influence martingale representation of the state operator (5) generically yields a solution of the master equation (1) we need to compute

$${{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}={{{{{{{\rm{d}}}}}}}}\,{{{{{{{\rm{E}}}}}}}}\left({\mu }_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right)$$
(30)

Differentiation commutes with the expectation value operation. Paths generated by stochastic differential equations have finite quadratic variation. We thus apply Itô lemma42

$${{{{{{{\rm{d}}}}}}}}({\mu }_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })=({{{{{{{\rm{d}}}}}}}}{\mu }_{t}){{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }+{\mu }_{t}{{{{{{{\rm{d}}}}}}}}({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })+({{{{{{{\rm{d}}}}}}}}{\mu }_{t}){{{{{{{\rm{d}}}}}}}}({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })$$
(31)

and observe that the explicit evaluation of

$${{{{{{{\rm{d}}}}}}}}({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })=({{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}){{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }+{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}({{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })+({{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t})({{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })$$
(32)

along the paths generated by (6) yields

$${{{{{{{\rm{d}}}}}}}}({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })= -\imath [{{{{{{{{\rm{H}}}}}}}}}_{t},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }]{{{{{{{\rm{d}}}}}}}}t\\ -\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }+{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }}{2}-{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right){{{{{{{\rm{d}}}}}}}}t\\ +\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell ,t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell ,t}^{{{{\dagger}}} }}{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}-{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right){{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}.$$
(33)

This last equation and the definitions of the counting process (7), (8) and influence martingale (9) differentials allow us to write

$${{{{{{{\rm{d}}}}}}}}({\mu }_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} })= \; ({{{{{{{\rm{d}}}}}}}}{\mu }_{t}){{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }-\imath \,{\mu }_{t}[{{{{{{{{\rm{H}}}}}}}}}_{t},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }]{{{{{{{\rm{d}}}}}}}}t\\ -{\mu }_{t}\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}\,\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }+{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }}{2}-{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right){{{{{{{\rm{d}}}}}}}}t\\ +{\mu }_{t}\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\frac{{{{\Gamma }}}_{\ell ,t}}{{{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,t}}\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }}{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}-{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right){{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}.$$
(34)

Once we take the expectation value, the telescopic property of conditional expectations see e.g., ref. 42 and the martingale property (12) guarantee that all terms proportional to increments of the influence martingale vanish. The remaining terms under expectation reduce to

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}= -\imath \,[{{{{{{{{\rm{H}}}}}}}}}_{t},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}]\\ -\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}\,\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}+{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }}{2}-{{{{{{{\rm{E}}}}}}}}{\mu }_{t}{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right)\\ +\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}{{{{{{{\rm{E}}}}}}}}{\mu }_{t}\left(\frac{{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }}{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}}-{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }\right){\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}\right\Vert }^{2}.$$
(35)

Straightforward algebra then allows us to recover (1).

Finally, we emphasize that the proof relies on the stochastic nature of the quadratic variation of the martingale component of the counting process. Generically, the quadratic variation of a martingale is a stochastic process42. At variance with the general case, Lévy’s characterization theorem proves that one of the defining properties of the Wiener process is self-averaging in mean square sense of the quadratic variation42. This fact hinders a straightforward extension of the above proof to quantum state diffusion15. Self-averaging of the quadratic variation is also a central element of the physics interpretation of quantum state diffusion. In fact, quantum state diffusion can be derived from the stochastic Schrödinger equation (6) in the singular limit of infinite number of detection events per unit of time2 and is, in this sense, adapted to describe a more restrictive class of physics contexts.

Linear stochastic differential equation

The Itô stochastic differential equation (6) preserves the Bloch hyper-sphere. On the hyper-sphere, we can look for solutions of the stochastic Schrödinger equation (6) of the form

$${{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}=\frac{{{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{t}}{\left\Vert {{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{t}\right\Vert }$$
(36)

The change of variables (36) maps (6) into the linear problem

$${{{{{{{\rm{d}}}}}}}}{{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{t}= -\imath \,{{{{{{{{\rm{H}}}}}}}}}_{t}{{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{t}\,{{{{{{{\rm{d}}}}}}}}t\\ -\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\left(\frac{{{{\Gamma }}}_{\ell ,t}}{2}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{\rm{d}}}}}}}}t-{{{{{{{\rm{d}}}}}}}}{\nu }_{\ell ,t}\left({{{{{{{{\rm{L}}}}}}}}}_{\ell }-{{{{{{{\rm{1}}}}}}}}\right)\right){{{{{{{{\boldsymbol{\varphi }}}}}}}}}_{t}$$
(37)

Once we know φt, we can use (36) to determine the state vector and the influence martingale. In particular, the influence martingale always admits the factorization

$${\mu }_{t}=\exp \left(\int\nolimits_{0}^{t}{{{{{{{\rm{d}}}}}}}}s\,\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}\left({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,s}-{{{\Gamma }}}_{\ell ,s}\right){\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{s}\right\Vert }^{2}\right)\,{\tilde{\mu }}_{t}$$
(38)

where \({\tilde{\mu }}_{t}\) is a pure jump process. This factorization is of use in numerical implementations. From (38) we readily see that negative values of the Γℓ,t’s exponentially enhance the contribution of the realization of the state vector to the expectation value.

Derivation of the stochastic Wittstock–Paulsen decomposition

Explicit expression of the probability measure

We describe sequences of detection events by first supposing that a fixed number n of jumps occurs in the time interval \(({t}_{{\mathfrak{0}}},t]\). Next, we suppose that a jump of type ℓi with ℓi taking values in \(\left\{1,2,\ldots ,{{{{{{{\mathscr{L}}}}}}}}\right\}\) occurs at time at si satisfying for i = 1, …, n the chain of inequalities

$$t \; > \; {s}_{n} \; > \; \ldots \; > \; {s}_{i}\, \; > \; \ldots \; > \; {s}_{{\mathfrak{1}}} \; > \; {s}_{{\mathfrak{0}}}={t}_{{\mathfrak{0}}}$$
(39)

An arbitrary sequence of detection events ω thus corresponds to a 2 n-tuple \({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\). The total number n of jumps ranges from zero to infinity. We refer to this mathematical description of events as “waiting time representation". On the Bloch hyper-sphere, all information about the dynamics of (6) in a time interval [s, t) during which no jump occurs is encapsulated in the Green function of the linear dynamics (37)

$${\dot{{{{{{{{\rm{G}}}}}}}}}}_{ts}=-\imath \,{{{{{{{{\rm{H}}}}}}}}}_{t}{{{{{{{{\rm{G}}}}}}}}}_{ts}-\frac{1}{2}\mathop{\sum }\limits_{\ell =1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{\Gamma }}}_{\ell ,t}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\rm{G}}}}}}}}}_{ts}+\delta (t-s)$$
(40)
$$\mathop{\lim }\limits_{t\searrow s}{{{{{{{{\rm{G}}}}}}}}}_{ts}={{{{{{{{\rm{1}}}}}}}}}_{{{{{{{{\mathcal{H}}}}}}}}}$$
(41)

Exactly repeating the same steps as in § 6.1 of ref. 2, we obtain the expression of the state vector conditional upon ω

$${{{{{{{\rm{E}}}}}}}}\left({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}| \omega ={\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n},{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{{t}_{{\mathfrak{0}}}}={{{{{{{\boldsymbol{z}}}}}}}}\right)=\frac{{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right){{{{{{{\boldsymbol{z}}}}}}}}}{\left\Vert {{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right){{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }.$$
(42)

By using the identity

$${e}^{-\int\nolimits_{s}^{t}{{{{{{{\rm{d}}}}}}}}u\mathop{\sum }\nolimits_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}{{{{{{{\mathscr{r}}}}}}}}_{\ell ,u}\frac{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}{\parallel {{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}{\parallel }^{2}}}={\left\Vert {{{{{{{{\rm{G}}}}}}}}}_{t{t}_{{\mathfrak{0}}}}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}{e}^{-\int\nolimits_{s}^{t}{{{{{{{\rm{d}}}}}}}}u\mathop{\sum }\nolimits_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,u}-{{{\Gamma }}}_{\ell ,u})\frac{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}{{\left\Vert {{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}}$$
(43)

we find that the multi-time probability density of the conditioning event is equal to

$${{{{{{{{\rm{p}}}}}}}}}_{t{t}_{{\mathfrak{0}}}}(\omega ={\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}| {{{{{{{{\boldsymbol{\psi }}}}}}}}}_{{t}_{{\mathfrak{0}}}}={{{{{{{\boldsymbol{z}}}}}}}})={\delta }_{n,0}\frac{{\left\Vert {{{{{{{{\rm{G}}}}}}}}}_{t{t}_{{\mathfrak{0}}}}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}{{{{{{{{{\mathscr{m}}}}}}}}}_{t{t}_{0}}({{{{{{{\boldsymbol{z}}}}}}}})}+\frac{(1-{\delta }_{n,0}){\left\Vert {{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right){{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}{{{{{{{{{\mathscr{m}}}}}}}}}_{t{s}_{n}}\left(\mathop{\prod }\nolimits_{j = 1}^{n}{{{{{{{{\rm{L}}}}}}}}}_{{\ell }_{j}}{{{{{{{{\rm{G}}}}}}}}}_{{s}_{j}{s}_{j-1}}{{{{{{{\boldsymbol{z}}}}}}}}\right)\ldots \,{{{{{{{{\mathscr{m}}}}}}}}}_{{s}_{1}{t}_{0}}({{{{{{{\boldsymbol{z}}}}}}}})}$$
(44)

In writing (42), (44) we introduced the tensor valued process

$${{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right):=\; {\delta }_{n,0}{{{{{{{{\rm{G}}}}}}}}}_{t{t}_{{\mathfrak{0}}}}+(1-{\delta }_{n,0}){{{{{{{{\rm{G}}}}}}}}}_{t{s}_{n}}\mathop{\prod }\limits_{i=1}^{n}\sqrt{{{{{{{{{\mathscr{r}}}}}}}}}_{{\ell }_{i},{s}_{i}}}{{{{{{{{\rm{L}}}}}}}}}_{{\ell }_{i}}{{{{{{{{\rm{G}}}}}}}}}_{{s}_{i}{s}_{i-1}}$$
(45)

and the scalar

$${{{{{{{{\mathscr{m}}}}}}}}}_{ts}({{{{{{{\boldsymbol{z}}}}}}}}):= {e}^{\int\nolimits_{s}^{t}{{{{{{{\rm{d}}}}}}}}u\mathop{\sum }\nolimits_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}({{{{{{{{\mathscr{r}}}}}}}}}_{\ell ,u}-{{{\Gamma }}}_{\ell ,u})\frac{{\left\Vert {{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}{{\left\Vert {{{{{{{{\rm{G}}}}}}}}}_{us}{{{{{{{\boldsymbol{z}}}}}}}}\right\Vert }^{2}}}$$
(46)

which is the value of the influence martingale if no jumps occur in the interval [s, t]. In order to neaten the notation in (42) and (44) and below, we omit to write the condition \({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{{t}_{{\mathfrak{0}}}}^{{{{\dagger}}} }={{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} }\) that fully specifies the initial data on the Bloch hyper-sphere.

If we now restrict the attention to the computation of quantum probabilities, we see that the product of the pure state operator specified by (42) times its probability (44) yields the stochastic dynamical map

$${{{{{{{{\mathbf{\Phi }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}[\omega ={\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}]({{{{{{{\boldsymbol{z}}}}}}}}{{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} })=\frac{{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{o}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right){{{{{{{\boldsymbol{z}}}}}}}}{{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} }{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{{{{\dagger}}} }\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right)}{{{{{{{{{\mathscr{m}}}}}}}}}_{t{s}_{n}}\left(\mathop{\prod }\nolimits_{j = 1}^{n}{{{{{{{{\rm{L}}}}}}}}}_{{\ell }_{j}}{{{{{{{{\rm{G}}}}}}}}}_{{s}_{j}{s}_{j-1}}{{{{{{{\boldsymbol{z}}}}}}}}\right)\ldots \,{{{{{{{{\mathscr{m}}}}}}}}}_{{s}_{1}{t}_{0}}({{{{{{{\boldsymbol{z}}}}}}}})}$$
(47)

satisfying by construction the unit trace condition

$$\int {\mathfrak{m}}({{{{{{{\rm{d}}}}}}}}\omega ){{{{{{{\rm{Tr}}}}}}}}{{{{{{{{\mathbf{\Phi }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}[\omega ]({{{{{{{\boldsymbol{z}}}}}}}}{{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} }):= \mathop{\sum }\limits_{n=0}^{\infty }\mathop{\prod }\limits_{i=1}^{n}\mathop{\sum }\limits_{{\ell }_{i}=1}^{{{{{{{{\mathscr{L}}}}}}}}}\int\nolimits_{{t}_{{\mathfrak{0}}}}^{t}{{{{{{{\rm{d}}}}}}}}{s}_{i}{{{{{{{{\rm{p}}}}}}}}}_{t{t}_{{\mathfrak{0}}}}(\omega ={\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n})=1.$$
(48)

Thus the dynamical map (47) takes the form of the generalized operator sum representation22,69. It differs from the Choi representation of a completely positive map (see e.g., ref. 23) in consequence of the non-linear dependence upon the initial state.

Stochastic Wittstock–Paulsen decomposition

Multiplying (47) by the influence martingale (38) occasions the cancellation of any non-linear dependence upon the initial data

$${{{{{{{\rm{E}}}}}}}}({\mu }_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{t}^{{{{\dagger}}} }| \omega ,{{{{{{{{\boldsymbol{\psi }}}}}}}}}_{{t}_{{\mathfrak{0}}}}={{{{{{{\boldsymbol{z}}}}}}}})={{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(+)}(\omega ){{{{{{{\boldsymbol{z}}}}}}}}{{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} }{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(+){{{\dagger}}} }(\omega )-{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(-)}(\omega ){{{{{{{\boldsymbol{z}}}}}}}}{{{{{{{{\boldsymbol{z}}}}}}}}}^{{{{\dagger}}} }{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(-){{{\dagger}}} }(\omega )$$
(49)

with

$${{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(\pm )}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right)={\left(\frac{\max (0,\pm \mathop{\prod }\limits_{k = 1}^{n}{{{\Gamma }}}_{{\ell }_{k},{s}_{k}})}{\mathop{\prod }\nolimits_{j = 1}^{n}{{{{{{{{\mathscr{r}}}}}}}}}_{{\ell }_{j},{s}_{j}}}\right)}^{1/2}{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t{t}_{{\mathfrak{0}}}}\left({\left\{{\ell }_{i},{s}_{i}\right\}}_{i = 1}^{n}\right)$$
(50)

We thus verify that (15) is indeed an expectation value over the difference of completely positive stochastic dynamical maps.

Derivation of the Redfield equation for a two-qubit system

We consider two non-interacting qubits in contact with an environment consisting of \({{{{{{{\mathscr{N}}}}}}}}\uparrow \infty\) bosonic oscillators

$$\hat{{{{{{{{\rm{H}}}}}}}}}=\mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}}=1}^{2}{\omega }^{({{{{{{{\mathscr{i}}}}}}}})}{\sigma }_{+}^{({{{{{{{\mathscr{i}}}}}}}})}{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})}+\mathop{\sum }\limits_{k=1}^{{{{{{{{\mathscr{N}}}}}}}}}\left({\epsilon }_{k}\,{{{{{{{{\rm{b}}}}}}}}}_{k}^{{{{\dagger}}} }{{{{{{{{\rm{b}}}}}}}}}_{k}+\mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}}=1}^{2}{g}_{k}\left({\sigma }_{+}^{({{{{{{{\mathscr{i}}}}}}}})}{{{{{{{{\rm{b}}}}}}}}}_{k}+{{{{{{{{\rm{b}}}}}}}}}_{k}^{{{{\dagger}}} }{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})}\right)\right)$$

Here \({\sigma }_{\pm }^{({{{{{{{\mathscr{i}}}}}}}})}\) is the lift to the tensor product Hilbert space of the ladder operators acting on the Hilbert space of individual qubits \({{{{{{{\mathscr{i}}}}}}}}=1,2\). Tracing out the environment yields the master equation59

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}=-\imath \mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}},{{{{{{{\mathscr{j}}}}}}}}=1}^{2}{{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}}\,[{\sigma }_{+}^{({{{{{{{\mathscr{j}}}}}}}})}{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}]+\mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}},{{{{{{{\mathscr{j}}}}}}}}=1}^{2}{{{{{{{{\rm{B}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}}\frac{\left[{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{\sigma }_{+}^{({{{{{{{\mathscr{j}}}}}}}})}\right]+\left[{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})}{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t},{\sigma }_{+}^{({{{{{{{\mathscr{j}}}}}}}})}\right]}{2}$$
(51)

where the \({{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}}\)’s and \({{{{{{{{\rm{B}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}}\)’s are respectively the components of the matrix

$${{{{{{{\rm{A}}}}}}}}=\left[\begin{array}{cc}\alpha &\alpha +\frac{\kappa }{2}-\imath \frac{{\gamma }_{2}\;-\;{\gamma }_{1}}{4}\\ \alpha +\frac{\kappa }{2}-\imath \frac{{\gamma }_{1}\;-\;{\gamma }_{2}}{4}&\alpha +\kappa \end{array}\right]$$
(52)

and

$${{{{{{{\rm{B}}}}}}}}=\frac{1}{2}\left[\begin{array}{cc}{\gamma }_{1}&\frac{{\gamma }_{1}\;+\;{\gamma }_{2}}{2}-\imath \,\varkappa \\ \frac{{\gamma }_{1}\;+\;{\gamma }_{2}}{2}+\imath \,\kappa &{\gamma }_{2}\end{array}\right]$$
(53)

The key observation is that the numbers γ1, γ2 are positive and α, ϰ are real. Thus, the matrix B is self-adjoint, and it is therefore unitarily equivalent to a real diagonal matrix

$${{{{{{{\rm{diag}}}}}}}}\;{{{{{{{\rm{B}}}}}}}}={{{{{{{{\rm{U}}}}}}}}}^{{{{\dagger}}} }\,{{{{{{{\rm{B}}}}}}}}\,{{{{{{{\rm{U}}}}}}}}=\left[\begin{array}{cc}{\lambda }_{1}&0\\ 0&{\lambda }_{2}\end{array}\right]$$
(54)

where for \({{{{{{{\mathscr{i}}}}}}}}=1,2\)

$${\lambda }_{{{{{{{{\mathscr{i}}}}}}}}}=\frac{{\gamma }_{1}+{\gamma }_{2}}{4}+{(-1)}^{i}\sqrt{\frac{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+2\,{\kappa }^{2}}{8}}$$
(55)

Upon defining the Lindblad operators

$${{{{{{{{\rm{L}}}}}}}}}_{{{{{{{{\mathscr{j}}}}}}}}}=\mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}}=1}^{2}{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})}\,{{{{{{{{\rm{U}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}},\,\,\,\,\,\,{{{{{{{\mathscr{j}}}}}}}}=1,2$$
(56)

and the self-adjoint operator

$${{{{{{{\rm{H}}}}}}}}=\mathop{\sum }\limits_{{{{{{{{\mathscr{i}}}}}}}},{{{{{{{\mathscr{j}}}}}}}}=1}^{2}{{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\mathscr{i}}}}}}}}{{{{{{{\mathscr{j}}}}}}}}}{\sigma }_{+}^{({{{{{{{\mathscr{j}}}}}}}})}{\sigma }_{-}^{({{{{{{{\mathscr{i}}}}}}}})}$$
(57)

we finally arrive at the master equation

$${\dot{{{{{{{{\boldsymbol{\rho }}}}}}}}}}_{t}=-\imath \,[{{{{{{{\rm{H}}}}}}}},{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}]+\mathop{\sum }\limits_{\ell =1}^{2}{\lambda }_{\ell }\frac{\left[{{{{{{{{\rm{L}}}}}}}}}_{\ell },{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }\right]+\left[{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t},{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }\right]}{2}$$
(58)

QuTiP implementation

For the numerics, we assign the Lindblad weights to be

$${{{\Gamma }}}_{1,t}=\gamma -12\exp (-2{t}^{3}){\sin }^{2}(15t)$$
(59)

and

$${{{\Gamma }}}_{\ell ,t}=\gamma \; > \; 0\,\,\,\,\,\ell =2,\ldots ,{{{{{{{\mathscr{N}}}}}}}}$$
(60)
$${{{\Gamma }}}_{\ell +{{{{{{{\mathscr{N}}}}}}}},t}=\delta \; > \; 0\,\,\,\,\,\,\ell =1,\ldots ,{{{{{{{\mathscr{N}}}}}}}}$$
(61)

In the time interval [0.2, 0.25] we choose a solution of the over-determined system

$${{{\Gamma }}}_{1,t}={{{{{{{{\mathscr{r}}}}}}}}}_{1,t}+{c}_{t}$$
(62)
$$\delta ={{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}+{c}_{t}$$
(63)

by defining

$${c}_{t}=-\frac{(1-{{{{{{{\rm{sign}}}}}}}}({{{\Gamma }}}_{1,t}))}{2}\frac{{{{\Gamma }}}_{1,t}-\delta /2}{{{{\Gamma }}}_{1,t}-\delta }\,{{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}},t}.$$
(64)

The choice of ct is merely based on the empirical observation that for the values of γ and δ we used, the resulting values of \({{{{{{{{\mathscr{r}}}}}}}}}_{1+{{{{{{{\mathscr{N}}}}}}}}}\), \({{{{{{{{\mathscr{r}}}}}}}}}_{1}\) are positive and efficiently handled by QuTiP.

Photo-current oscillations

One way to conceptualize time-convolutionless perturbation theory is as an avenue to implement at any order in the system environment coupling constant a Markov approximation in the derivation of the master equation. The leading order corresponds to the weak coupling approximation. In that case, the Lindblad operators \({\left\{{{{{{{{{\rm{L}}}}}}}}}_{\ell }\right\}}_{\ell = 1}^{{{{{{{{\mathscr{L}}}}}}}}}\) are obtained as eigenoperators of the unperturbed isolated system Hamiltonian \({{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{0}}}\)2:

$$\left[{{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{0}}},{{{{{{{{\rm{L}}}}}}}}}_{\ell }\right]={\epsilon }_{\ell }{{{{{{{{\rm{L}}}}}}}}}_{\ell }$$
(65)

for some ϵl’s, taking positive and negative values. We straightforwardly verify that

$$\left[{{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{0}}},{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{{\rm{L}}}}}}}}}_{\ell }\right]=0$$
(66)

immediately follows. In general, \({{{{{{{{\rm{H}}}}}}}}}_{{\mathfrak{0}}}\) differs from the Hamilton operator Ht in (1) by Lamb shift corrections. It is then reasonable to surmise that higher order corrections determined by time convolutionless perturbation theory only affect the intensity of the Lamb shift and the values of weight functions Γℓ,t of the Lindblad operators in (1) see e.g., ref. 70. Under these hypotheses a straightforward calculation using

$${{{{{{{\rm{d}}}}}}}}{{{{{{{\rm{E}}}}}}}}({\mu }_{t}{\nu }_{\ell ,t})={{{\Gamma }}}_{\ell ,t}{{{{{{{\rm{Tr}}}}}}}}{{{{{{{{\rm{L}}}}}}}}}_{\ell }{{{{{{{{\boldsymbol{\rho }}}}}}}}}_{t}{{{{{{{{\rm{L}}}}}}}}}_{\ell }^{{{{\dagger}}} }{{{{{{{\rm{d}}}}}}}}t$$
(67)

allows us to derive the energy balance equation (29).

Measurement interpretation

The mathematical notion of “instrument” \({{{{{{{\mathcal{I}}}}}}}}\) provides the description of quantum measurement adapted to quantum trajectory theory see e.g., ref. 7. An instrument is a map from a classical probability space \(({{\Omega }},{{{{{{{\mathcal{F}}}}}}}},{{{{{{{\rm{d}}}}}}}}P)\)42 to the space of bounded operators acting on a Hilbert space \({{{{{{{\mathcal{H}}}}}}}}\) that for any pre-measurement state operator X satisfies

$${{{{{{{{\mathcal{I}}}}}}}}}_{t}(F)[{{{{{{{\boldsymbol{X}}}}}}}}]=\mathop{\sum}\limits_{k}{\int}_{F}{{{{{{{\rm{d}}}}}}}}P(\omega ){{{{{{{{\mathcal{V}}}}}}}}}_{t,k}(\omega ){{{{{{{\boldsymbol{X}}}}}}}}{{{{{{{{\mathcal{V}}}}}}}}}_{t,k}^{{{{\dagger}}} }(\omega )$$
(68a)
$$\mathop{\sum}\limits_{k}{\int}_{F}{{{{{{{\rm{d}}}}}}}}P(\omega ){{{{{{{{\mathcal{V}}}}}}}}}_{t,k}^{{{{\dagger}}} }(\omega ){{{{{{{{\mathcal{V}}}}}}}}}_{t,k}(\omega )={{{{{{{{\rm{1}}}}}}}}}_{{{{{{{{\mathcal{H}}}}}}}}}$$
(68b)

Here dP is a classical probability measure, F ⊆ Ω is an event in the σ-algebra \({{{{{{{\mathcal{F}}}}}}}}\) describing all possible outcomes from the sample space Ω and \({{{{{{{{\mathcal{V}}}}}}}}}_{t,k}(\omega )\) are operators acting on \({{{{{{{\mathcal{H}}}}}}}}\). In (68a) we regard the instrument as a function of the time t.

In order to make contact with quantum trajectories unraveling a completely bounded state operator, we interpret \({{{{{{{\mathcal{H}}}}}}}}\) as an embedding Hilbert space \({{{{{{{\mathcal{H}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{E}\,\otimes \,{{{{{{{{\mathcal{H}}}}}}}}}_{S}\) and X = π(ρ0) as a representation of the initial state operator of the system onto \({{{{{{{\mathcal{H}}}}}}}}\). Finally let \({\left\{{{{{{{{{\rm{E}}}}}}}}}_{ij}\right\}}_{i,j = 1}^{\dim {{{{{{{{\mathcal{H}}}}}}}}}_{E}}\) be the canonical basis of the space of operators acting on \({{{{{{{{\mathcal{H}}}}}}}}}_{E}\). We assume \(\dim {{{{{{{{\mathcal{H}}}}}}}}}_{E} \; < \; \infty\) and recall that \({{{{{{{{\rm{E}}}}}}}}}_{ij}={{{{{{{{\boldsymbol{e}}}}}}}}}_{i}{{{{{{{{\boldsymbol{e}}}}}}}}}_{j}^{{{{\dagger}}} }\) where \({\left\{{{{{{{{{\boldsymbol{e}}}}}}}}}_{i}\right\}}_{i = 1}^{\dim {{{{{{{{\mathcal{H}}}}}}}}}_{E}}\) is the canonical basis of \({{{{{{{{\mathcal{H}}}}}}}}}_{E}\) itself. Let now O be a self-adjoint operator acting on \({{{{{{{{\mathcal{H}}}}}}}}}_{S}\). General results in linear operator algebra23 ensure the existence of an instrument such that for i ≠ j we can write

$${{{{{{{{\rm{Tr}}}}}}}}}_{{{{{{{{{\mathcal{H}}}}}}}}}_{S}}{{{{{{{\rm{O}}}}}}}}\left({{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(+)}(\omega ){{{{{{{{\boldsymbol{\rho }}}}}}}}}_{0}{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(+){{{\dagger}}} }(\omega )-{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(-)}(\omega ){{{{{{{{\boldsymbol{\rho }}}}}}}}}_{0}{{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(-){{{\dagger}}} }(\omega )\right)\\ =\mathop{\sum}\limits_{k}{{{{{{{{\rm{Tr}}}}}}}}}_{{{{{{{{\mathcal{H}}}}}}}}}\left(\frac{{{{{{{{{\rm{E}}}}}}}}}_{ij}+{{{{{{{{\rm{E}}}}}}}}}_{i\,j}^{{{{\dagger}}} }}{2}\,\otimes \,{{{{{{{\rm{O}}}}}}}}\,{{{{{{{{\mathcal{V}}}}}}}}}_{t{t}_{{\mathfrak{0}}},k}(\omega ){{{{{{{\boldsymbol{X}}}}}}}}{{{{{{{{\mathcal{V}}}}}}}}}_{t\,{t}_{{\mathfrak{0}}},k}^{{{{\dagger}}} }(\omega )\right)$$
(69)

for \({{{{{{{{\mathbf{\Lambda }}}}}}}}}_{t\,{t}_{{\mathfrak{0}}}}^{(\pm )}\) defined in (50) and some \({{{{{{{{\mathcal{V}}}}}}}}}_{t{t}_{{\mathfrak{0}}},k}(\omega )\). We refer to ref. 37 for an explicit construction of an embedding representation.