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arXiv:2605.01411 · Journal of Physics A Mathematical and Theoretical

Quantum jump trajectories, hybrid systems, non-Hermitian evolutions, quantum/classical walks

Alberto Barchielli

quant-phmath-phSubmitted May 2, 2026• Announced May 5, 2026• 0 citations
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Abstract

Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...

AI Summary

The paper asks how jump-type stochastic master equations can provide a common description of continuous monitoring, quantum/classical hybrid dynamics, non-Hermitian evolution, renewal processes and continuous-time open quantum walks. It formulates linear and normalized nonlinear SMEs with continuously or discretely labelled jumps, allowing feedback and additional classical noise. A change from a reference probability measure to the physical measure produces state-dependent jump intensities; instruments and POVMs connect the construction to quantum measurement theory. Finite jump records, called “typical trajectories,” give recursive solutions by alternating completely positive no-jump propagators and jump maps. Their traces yield exclusive probability densities encoding jump times, labels and counting statistics. For deterministic coefficients, the next-jump survival probability is $S(t)=\operatorname{Tr}\{\mathcal S(t,0)[\rho_0]\}$, with density $-S'(t)$ and a possible atom at infinite waiting time. For a bounded non-Hermitian Hamiltonian, the construction requires $R=c\openone+i(H_{\rm eff}-H_{\rm eff}^{\dagger})\geq0$. Normalized no-jump dynamics is independent of $c$, but survival statistics are not. Explicit two-level calculations cover exceptional points and oscillatory waiting-time densities. Further constructions recover prescribed renewal-time laws and Lindblad rate equations as hybrid dynamics; an Erlang example connects mean-state distinguishability revivals to classical jump-record probabilities. The derivations use stochastic calculus, propagator expansions and analytic qubit examples. The framework assumes bounded operators and finite total reference jump intensity, excluding infinite-activity jumps. Relaxing these assumptions is not developed. With random coefficients, the mean state generally has no closed evolution equation, and partial-observation conditional states generally lack explicit formulas.

No sign of AI-generated textIncrementalUnclear novelty

Central claim may not hold: The claimed global physical probability need not exist on the stipulated Q-completed filtration: a scalar jump operator changing a Poisson rate produces finite-time laws incompatible with an initial-sigma-algebra null event. Equation (survPupm) reverses the exponential norm factors of the two eigencomponents, giving incorrect survival rates when the eigenvalues have nonzero imaginary parts. Equation group (refex) incorrectly substitutes the loss parameters into tilde c: the earlier definition requires c+Γ_1+Γ_0 rather than c−(Γ_1+Γ_0)/2.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Journal reference
J. Phys. A: Math. Theor. 59 (2026) 345305
DOI
10.1088/1751-8121/ae96e1
Comments
54 pages

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Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...
The paper asks how jump-type stochastic master equations can provide a common description of continuous monitoring, quantum/classical hybrid dynamics, non-Hermitian evolution, renewal processes and continuous-time open quantum walks. It formulates linear and normalized nonlinear SMEs with continuously or discretely labelled jumps, allowing feedback and additional classical noise. A change from a reference probability measure to the physical measure produces state-dependent jump intensities; instruments and POVMs connect the construction to quantum measurement theory.
Finite jump records, called “typical trajectories,” give recursive solutions by alternating completely positive no-jump propagators and jump maps. Their traces yield exclusive probability densities encoding jump times, labels and counting statistics. For deterministic coefficients, the next-jump survival probability is S(t)=Tr⁡{S(t,0)[ρ0]}S(t)=\operatorname{Tr}\{\mathcal S(t,0)[\rho_0]\}S(t)=Tr{S(t,0)[ρ0​]}, with density −S′(t)-S'(t)−S′(t) and a possible atom at infinite waiting time.
For a bounded non-Hermitian Hamiltonian, the construction requires R=c\openone+i(Heff−Heff†)≥0R=c\openone+i(H_{\rm eff}-H_{\rm eff}^{\dagger})\geq0R=c\openone+i(Heff​−Heff†​)≥0. Normalized no-jump dynamics is independent of ccc, but survival statistics are not. Explicit two-level calculations cover exceptional points and oscillatory waiting-time densities. Further constructions recover prescribed renewal-time laws and Lindblad rate equations as hybrid dynamics; an Erlang example connects mean-state distinguishability revivals to classical jump-record probabilities.
The derivations use stochastic calculus, propagator expansions and analytic qubit examples. The framework assumes bounded operators and finite total reference jump intensity, excluding infinite-activity jumps. Relaxing these assumptions is not developed. With random coefficients, the mean state generally has no closed evolution equation, and partial-observation conditional states generally lack explicit formulas.
Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.
We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(psi,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring autonomous second-moment evolution and trace preservation, together with a full-column-rank quantum-classical noise correlation matrix, forces the ensemble dynamics into hybrid Lindblad form. The same result also determines the general class of diffusive stochastic unravelings compatible with this hybrid evolution. Notably, this construction makes positivity immediate and does not take complete positivity or an underlying unitary quantum dilation as assumptions, but instead relies on the presence of a sufficiently coupled classical sector. We formulate quantum measurement, filtering and smoothing of hybrid quantum-classical dynamics as standard Bayesian inference and show how to correlation between the quantum and classical noise fully explains the differences between quantum smoothing and smoothing in classical time series models. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories. These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment.
We show that all nonrelativistic quantum processes, whether open or closed, are either unitary or probabilistic unitary, i.e., probabilistic combinations of unitary evolutions. This means that, for open quantum systems, their continuous dynamics can always be described by the Lindblad master equation with all jump operators being unitary. We call this formalism the probabilistic unitary formulation of open quantum system dynamics. This formalism is shown to be exact under all cases, and does not rely on any assumptions other than the continuity and differentiability of the density matrix. Moreover, it requires as few as $d\ensuremath{-}1$ jump operators, instead of ${d}^{2}\ensuremath{-}1$, to describe the open dynamics in the most general case, where ddd is the dimension of Hilbert space of the system. Importantly, different from the conventional Lindblad master equation, this formalism is state dependent, meaning that the Hamiltonian, jump operators, and rates in general all depend on the current state of the density matrix. Hence one needs to know the explicit expression of the density matrix in order to write down the probabilistic unitary master equation explicitly. Experimentally, the formalism provides a scheme to control a quantum state to evolve along designed nonunitary quantum trajectories, and can be potentially useful in quantum computing and quantum control scenes since only unitary resources are needed for implementation.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
Lindbladian dissipative processes can be determined by stochastic samplings of individual trajectories, whose non-Hermitian evolution is perturbed by quantum jumps. The conditional dynamics of the jump events is fundamental to access many relevant statistical quantities of the dissipative dynamics. In this work, we mostly focus on the conditional no-jump evolution of noninteracting chains, including partial missed detection of jumps. As a main result, for Gaussian-preserving Lindblad operators, the conditional dynamics is captured by a nonlinear Riccati-type differential equation for the correlation matrix. As an application, we study the no-jump probability and the waiting-time distributions for chains of hopping particles and double quantum dots.