Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.08735 · Physical Review A

Restricted Monte Carlo wave-function method and Lindblad equation for identifying entangling open-quantum-system dynamics

Laura Ares, Julien Pinske, Benjamin Hinrichs, Martin Kolb, Jan Sperling

Published Dec 12, 2025• 5 citations
arXivHTMLPDF

Abstract

We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.

DOI
10.1103/hcj7-8zlg

Related papers

Separability Lindblad equation for dynamical open-system entanglement

Julien Pinske, Laura Ares, Benjamin Hinrichs +2 more

Providing entanglement for the design of quantum technologies in the presence of noise constitutes today's main challenge in quantum information science. A framework is required that assesses the build-up of entanglement in realistic settings. In this work, we put forth a new class of nonlinear quantum master equations in Lindblad form that unambiguously identify dynamical entanglement in open quantum systems via deviations from a separable evolution. This separability Lindblad equation restricts quantum trajectories to classically correlated states only. Unlike many conventional approaches, here the entangling capabilities of a process are not characterized by input-output relations, but separability is imposed at each instant of time. We solve these equations for crucial examples, thereby quantifying the dynamical impact of entanglement in non-equilibrium scenarios. Our results allow to benchmark the engineering of entangled states through dissipation. The separability Lindblad equation provides a unique path to characterizing quantum correlations caused by arbitrary system-bath interactions, specifically tailored for the noisy intermediate-scale quantum era.

arXiv:2412.08724•Physical Review A•Dec 3, 2025•5 citations

Stochastic wave-function unravelling of the generalized Lindblad equation using correlated states

Mervlyn Moodley, S Paul, T Nsio Nzundu

We perform a stochastic wave-function unravelling of the generalized Lindblad master equation using correlated states, a combination of the system state vectors and the environment population. The time-convolutionless projection operator method using correlated projection superoperators is applied to a two-state system, a qubit, that is coupled to an environment consisting of two energy bands which are both populated. These results are compared to the data obtained from Monte Carlo wave-function simulations based on the unravelling of the master equation. We also show a typical quantum trajectory and the average time evolution of the state vector on the Bloch sphere.

Physica Scripta•Mar 7, 2012•1 citation

Quantum Trajectories for Time-Local Non-Lindblad Master Equations

Tobias Becker, Ché Netzer, André Eckardt

For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.

arXiv:2306.14876•Physical Review Letters•Oct 17, 2023•11 citations

Stochastic dynamics of open quantum systems: Derivation of the differential Chapman-Kolmogorov equation

Heinz‐Peter Breuer, Francesco Petruccione

A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1995•23 citations

On the Unraveling of Open Quantum Dynamics

Brecht Donvil, Paolo Muratore-Ginanneschi

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.

arXiv:2309.13408•Open Systems & Information Dynamics•Sep 1, 2023•5 citations

Separable and Inseparable Quantum Trajectories

Jan Sperling, Ian A. Walmsley

The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schrödinger-type equations allow for comparing the trajectories in a separable configuration with the actual behavior of the system and, thereby, identifying inseparable and time-dependent quantum properties. As an example, we study bipartite discrete- and continuous-variable interacting systems. The generalization of our developed technique to multipartite scenarios is also provided.

arXiv:1708.04421•Physical Review Letters•Oct 24, 2017•9 citations
We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.
Providing entanglement for the design of quantum technologies in the presence of noise constitutes today's main challenge in quantum information science. A framework is required that assesses the build-up of entanglement in realistic settings. In this work, we put forth a new class of nonlinear quantum master equations in Lindblad form that unambiguously identify dynamical entanglement in open quantum systems via deviations from a separable evolution. This separability Lindblad equation restricts quantum trajectories to classically correlated states only. Unlike many conventional approaches, here the entangling capabilities of a process are not characterized by input-output relations, but separability is imposed at each instant of time. We solve these equations for crucial examples, thereby quantifying the dynamical impact of entanglement in non-equilibrium scenarios. Our results allow to benchmark the engineering of entangled states through dissipation. The separability Lindblad equation provides a unique path to characterizing quantum correlations caused by arbitrary system-bath interactions, specifically tailored for the noisy intermediate-scale quantum era.
We perform a stochastic wave-function unravelling of the generalized Lindblad master equation using correlated states, a combination of the system state vectors and the environment population. The time-convolutionless projection operator method using correlated projection superoperators is applied to a two-state system, a qubit, that is coupled to an environment consisting of two energy bands which are both populated. These results are compared to the data obtained from Monte Carlo wave-function simulations based on the unravelling of the master equation. We also show a typical quantum trajectory and the average time evolution of the state vector on the Bloch sphere.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
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.
The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schrödinger-type equations allow for comparing the trajectories in a separable configuration with the actual behavior of the system and, thereby, identifying inseparable and time-dependent quantum properties. As an example, we study bipartite discrete- and continuous-variable interacting systems. The generalization of our developed technique to multipartite scenarios is also provided.