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arXiv:2607.09651

Local Determinacy of Quantum Master Equations and a Mechanical Interpretation of the Multi-Mode Jaynes-Cummings and Central Spin Models

Jaeha Lee, Miku Ishizaki

quant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
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Abstract

We propose a novel formulation of master equations for open systems wherein the evolution of a state is determined solely by its local behaviour at any point of time. Specifically, our formulation allows for a local interpretation of the workings of beyond-Markovian dynamics as opposed to the more common conception that non-Markovian state evolution is affected by its cumulative past history. Quite interestingly, local determinacy is found prevalent in quantum dynamics. We illustrate the advantages of our coordinate-free formulation with exact analyses on two physically relevant models.

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5 pages

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We propose a novel formulation of master equations for open systems wherein the evolution of a state is determined solely by its local behaviour at any point of time. Specifically, our formulation allows for a local interpretation of the workings of beyond-Markovian dynamics as opposed to the more common conception that non-Markovian state evolution is affected by its cumulative past history. Quite interestingly, local determinacy is found prevalent in quantum dynamics. We illustrate the advantages of our coordinate-free formulation with exact analyses on two physically relevant models.
Master equations describing open quantum dynamics are typically first-order differential equations. When such dynamics brings the trajectories in state space of more than one initial state to the same point at finite instants in time, the generator of the corresponding master equation becomes singular while the dynamical map becomes noninvertible. The first-order, time-local, homogeneous master equations then fail to describe the dynamics beyond the singular point. Retaining time locality in the master equation necessitates a reformulation in terms of higher-order differential equations. We formulate a method to eliminate the divergent behavior of the generator by using a combination of higher-order derivatives of the generator with suitable weights and illustrate it with several examples. We also present a detailed study of the central spin model and we propose the average rate of information inflow in non-Markovian processes as a quantity that captures a different aspect of non-Markovian dynamics.
Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics.
In this review we discuss recent progress in the theory of open quantum systems based on non-MarkovianNon-Markovian quantum state diffusionQuantum state diffusion equation and master equationsMaster equation. In particular, we show that an exact master equation for an open quantum system consisting of a few qubits can be explicitly constructed by using the corresponding non-Markovian quantum state diffusion equation. The exact master equation arises naturally from the quantum decoherence dynamics of qubit systems collectively interacting with a colored noise. We illustrate our general theoretical formalism by the explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This exact qubit master equation accurately characterizes the time evolution of the qubit system in various parameter domains, and paves the way for investigation of the memory effect of an open quantum system in a non-Markovian regime without any approximation.
A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum-state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation, and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative NNN-qubit system can be derived explicitly from the statistical average of the corresponding non-Markovian quantum trajectories. We illustrated our general formulation by an explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This multiple qubit master equation offers an accurate time evolution of quantum systems in various domains, and paves the way to investigate the memory effect of an open system in a non-Markovian regime without any approximation.
Non-Markovian processes may arise in physics due to memory effects of environmental degrees of freedom. For quantum non-Markovianity, it is an ongoing debate to clarify whether such memory effects have a verifiable quantum origin, or whether they might equally be modeled by a classical memory. In this contribution, we propose a criterion to test locally for a truly quantum memory. The approach is agnostic with respect to the environment, as it solely depends on the local dynamics of the system of interest. Experimental realizations are particularly easy, as only single-time measurements on the system itself have to be performed. We study memory in a variety of physically motivated examples, both for a time-discrete case, and for time-continuous dynamics. For the latter, we are able to provide an interesting class of non-Markovian master equations with classical memory that allows for a physically measurable quantum trajectory representation.