Efficient simulation of Markovian open quantum systems remains a central challenge because the density-matrix description grows exponentially with system size. Quantum trajectory methods provide an alternative by replacing mixed-state evolution with stochastic pure-state realizations. Here we investigate this framework for normal Lindblad generators, whose orthogonal eigenoperator decomposition precludes transient amplification. By decomposing the Lindbladian into deterministic smooth and stochastic jump contributions, we derive an exact steady-state balance relation that identifies the interplay between these processes as the mechanism underlying Liouvillian normality. We further show that normal Lindbladians exclude exceptional points and that, although individual quantum trajectories generally exhibit stochastic coupling between Liouvillian eigenmodes, these couplings cancel upon ensemble averaging, recovering independent orthogonal relaxation modes. These results provide a trajectory-level interpretation of Liouvillian normality and clarify how a global property of the Lindblad generator is realized through stochastic quantum dynamics.
Efficient simulation of Markovian open quantum systems remains a central challenge because the density-matrix description grows exponentially with system size. Quantum trajectory methods provide an alternative by replacing mixed-state evolution with stochastic pure-state realizations. Here we investigate this framework for normal Lindblad generators, whose orthogonal eigenoperator decomposition precludes transient amplification. By decomposing the Lindbladian into deterministic smooth and stochastic jump contributions, we derive an exact steady-state balance relation that identifies the interplay between these processes as the mechanism underlying Liouvillian normality. We further show that normal Lindbladians exclude exceptional points and that, although individual quantum trajectories generally exhibit stochastic coupling between Liouvillian eigenmodes, these couplings cancel upon ensemble averaging, recovering independent orthogonal relaxation modes. These results provide a trajectory-level interpretation of Liouvillian normality and clarify how a global property of the Lindblad generator is realized through stochastic quantum dynamics.
Abstract The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schrödinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schrödinger and the quantum-jump dynamics.
Understanding the structure and stability of open quantum dynamics is increasingly important for both fundamental studies of nonequilibrium quantum systems and the development of quantum simulation algorithms. In this work, we introduce a structural framework for Markovian open quantum systems that characterizes Lindbladian generators in terms of two scalar quantities: the dissipative strength and the non-normality. We show that normal generators admit an exact decoupling between dissipative and norm-preserving dynamics, leading to purely exponential behavior governed by the dissipative scale. In contrast, non-normality is an intrinsically dissipative feature: it vanishes in the absence of dissipation but is not implied by it. Moreover, it is structurally constrained by the interplay between the Hermitian and anti-Hermitian components of the generator. For generic Markovian open quantum systems, we identify parametric regimes controlled by a dimensionless ratio between non-normality and dissipative strength, governing the onset of transient amplification. These structural features have direct implications for quantum simulation. While Hamiltonian and normal dissipative dynamics exhibit stable evolution with standard scaling behavior, non-normal generators can induce transient growth that amplifies numerical errors and increases simulation cost. Our results provide a unified generator-level perspective on irreversibility, stability, and quantum simulation of open quantum systems.
We develop an approach for understanding the dynamics of open quantum systems by analyzing individual quantum trajectories in the eigenbasis of the Liouvillian superoperator. From trajectory-eigenstate overlaps, we construct a quasiprobability distribution that characterizes the degree of localization of the trajectories in the Liouvillian eigenbasis. Contrary to the common wisdom that late-time dynamics are governed solely by the steady state and the slowest-decaying modes, we show that trajectories can remain well spread over transient eigenstates deep within the bulk of the Liouvillian spectrum even at late times. We demonstrate this explicitly using numerical simulations of interacting spin chains and bosonic systems. Moreover, we find that the delocalization of the trajectory strongly correlates with the purity of the trajectory-averaged steady state, establishing a further link between the trajectory and ensemble pictures of open quantum dynamics.
We derive the L-mean-field Ehrenfest (MFE) method to incorporate Lindblad jump operator dynamics into the MFE approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories that exactly reproduce Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.
The Lindblad master equation (LME) describing the Markovian dynamics of the quantum open system can be understood as the evolution of the effective non-Hermitian Hamiltonian balanced with random quantum jumps. Here we investigate the balance-breaking dynamics by partly eliminating jumps from postselection experiments. To describe this dynamics, a nonlinear Lindblad master equation (NLME) is derived from the quantum trajectory method. However, the NLME shows significant advantages in analytical analysis over the quantum trajectory method. Using the NLME, we classify the dynamics into two classes. In the trivial class, the process of reducing jumps is completely equivalent to weakening the coupling from the environment. In contrast, the nontrivial class presents more complex dynamics. We study a prototypical model within this class and demonstrate the existence of the postselected skin effect whose steady state is characterized by the accumulation of particles on one side. The steady-state distribution can be fitted by a scale-invariant tanh function which is different from the uniform distribution of LME. Furthermore, the NLME can give a reasonable framework for studying the interplay and competition between the non-Hermitian Hamiltonians and dissipative terms. We show this by capturing the characteristics of the trajectory-averaged entanglement entropy influenced by non-Hermitian skin effect and Zeno effect in the model of postselected skin effect.
The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. We demonstrate the efficiency of our method on several examples of coupled bosonic systems.