By using a condition of average trace preservation we rederive a general class of non-Markovian Gaussian diffusive unravelings [L. Di'osi and L. Ferialdi, Phys. Rev. Lett. 113, 200403 (2014)], here valid for arbitrary non-Hermitian system operators and noise correlations. The conditions under which the generalized stochastic Schr"odinger equation has the same symmetry properties (invariance under unitary changes of operator base) as a microscopic system-bath Hamiltonian dynamics are determined. While the standard quantum diffusion model (standard noise correlations) always shares the same invariance symmetry, the generalized stochastic dynamics can be mapped with an arbitrary bosonic environment only if some specific correlation constraints are fulfilled. These features are analyzed for different non-Markovian unravelings equivalent on average. Results based on quantum measurement theory that leads to specific cases of the generalized dynamics [J. Gambetta and H. M. Wiseman, Phys. Rev. A 66, 012108 (2002)] are studied from the perspective of the present analysis.
We derive a family of Gaussian non-Markovian stochastic Schrödinger equations for the dynamics of open quantum systems. The different unravelings correspond to different choices of squeezed coherent states, reflecting different measurement schemes on the environment. Consequently, we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the Gaussian process. By construction, the reduced dynamics of the open system does not depend on the squeezing parameters. They determine the non-Hermitian Gaussian correlation, a wide range of which are compatible with the Markov limit. We demonstrate the versatility of our results for quantum information tasks in the non-Markovian regime. In particular, by optimizing the squeezing parameters, we can tailor unravelings for improving entanglement bounds or for environment-assisted entanglement protection.
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows us to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative non-Markovian collapse models.
General open quantum systems display memory features, their master equations are non-Markovian. We show that the subclass of Gaussian non-Markovian open system dynamics is tractable in a depth similar to the Markovian class. The structure of master equations exhibits a transparent generalization of the Lindblad structure. We find and parametrize the class of stochastic Schrödinger equations that unravel a given master equation, such a class was previously known for Markovian systems only. We show that particular non-Markovian unravelings known in the literature are special cases of our class.
Quantum measurements and the associated state changes are properly described in the language of instruments.We investigate the properties of a time continuous family of instruments associated with the recently introduced class of general Gaussian non-Markovian stochastic Schrödinger equations.In this article we find that when the covariance matrix for the Gaussian noise satisfies a particular δ-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator.This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions.Remarkably, due to quantum memory effects the reduced state, in general, does not obey a closed form master equation while the unraveling has a time continuous measurement interpretation.
The master equation for the state of an open quantum system can be unravelled into stochastic trajectories described by a stochastic master equation. Such stochastic differential equations can be interpreted as an update formula for the system state conditioned on results obtained from monitoring the bath. So far only one parametrization (mathematical representation) for arbitrary diffusive unravellings (quantum trajectories arising from monitorings with Gaussian white noise) of a system described by a master equation with
L Lindblad terms has been found [H. M. Wiseman and A. C. Doherty, Phys. Rev. Lett. 94, 070405 (2005)]. This parametrization, which we call the
U representation parameterizes diffusive unravellings by
L2+2L real numbers, arranged in a matrix
U subject to three constraints. In this paper we investigate alternative parametrizations of diffusive measurements. We find, rather surprisingly, the description of diffusive unravellings can be unified by a single equation for a nonsquare complex matrix
M if one is willing to allow for some redundancy by lifting the number of real parameters necessary from
L2+2L to
3L2+L. We call this parametrization the
M representation. Both the
M representation and
U representation lack a physical picture of what the measurement should look like. We thus propose another parametrization, the
B representation that details how the measurement is implemented in terms of beam-splitters, phase shifters, and homodyne detectors. Relations between the different representations are derived.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.