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arXiv:2510.05472

Quantum Regression Theory and Efficient Computation of Response Functions for Non-Markovian Open Systems

Xiantao Li, Chunhao Wang

Published Oct 7, 2025• 0 citations

Abstract

Linear response functions are a cornerstone concept in physics as they enable efficient estimation of many dynamical properties. In addition to predicting dynamics of observables under perturbations without resimulating the system, these response functions lead to electric conductivity, magnetic susceptibility, dielectric constants, etc. Estimating two-time correlation functions is a key ingredient of measuring linear response functions. However, for open quantum systems, simulating the reduced density operator with a quantum master equation only yields \emph{one-point} observables and is insufficient for this task. In this paper, we develop a memoryless, system-only formulation of two-point correlations for open quantum systems that extends the standard quantum regression theorem (QRT) beyond the Markov limit. We further incorporate the spectral property of the bath and express the time propagators in the response function as the memoryless generators in Lindblad-type forms. The resulting expressions recast the total response function into evolutions generated by time-dependent Hamiltonian and Lindblad primitives together with the more challenging propagation of commutators and anti-commutators. In addition to the derivation of the new QRT, we present quantum algorithms for these primitives and obtain an estimator for two-time correlations whose cost scales poly-logarithmically in the system dimension and $1/ε^{1.25}$ in the target accuracy $ε$. The framework removes the separability (Born-Markov) assumption and offers a pathway to efficient computation of nonequilibrium properties from open quantum systems.

DOI: 10.48550/arxiv.2510.05472

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Linear response functions are a cornerstone concept in physics as they enable efficient estimation of many dynamical properties. In addition to predicting dynamics of observables under perturbations without resimulating the system, these response functions lead to electric conductivity, magnetic susceptibility, dielectric constants, etc. Estimating two-time correlation functions is a key ingredient of measuring linear response functions. However, for open quantum systems, simulating the reduced density operator with a quantum master equation only yields \emph{one-point} observables and is insufficient for this task. In this paper, we develop a memoryless, system-only formulation of two-point correlations for open quantum systems that extends the standard quantum regression theorem (QRT) beyond the Markov limit. We further incorporate the spectral property of the bath and express the time propagators in the response function as the memoryless generators in Lindblad-type forms. The resulting expressions recast the total response function into evolutions generated by time-dependent Hamiltonian and Lindblad primitives together with the more challenging propagation of commutators and anti-commutators. In addition to the derivation of the new QRT, we present quantum algorithms for these primitives and obtain an estimator for two-time correlations whose cost scales poly-logarithmically in the system dimension and

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in the target accuracy

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A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.

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Temporal correlations are fundamental in quantum physics, yet their computation is often challenging. The regression theorem (or hypothesis) serves as a key tool in this context, offering a seemingly straightforward approach. However, it fails for systems strongly coupled to their surroundings, where memory effects become significant. Here, we extend the analysis of temporal correlations beyond the regression theorem, revealing what can be learned about open quantum systems when this hypothesis fails. We introduce robust, operationally meaningful methods to explore how the breakdown of the regression hypothesis can uncover fundamental quantum features of non-Markovian open systems, including entanglement, coherence, and quantum memory, namely, the fundamental impossibility of simulating memory in non-Markov processes with classical feedback mechanisms. Finally, we demonstrate how these quantum features are linked to microscopic properties such as the bath spectral density and heat flow.

Experiments performed on quantum systems often measure multitime correlation functions. When quantum systems are weakly coupled to their environment, the time evolution of such correlation functions can be reduced to that of the reduced density matrix by the quantum regression theorem (QRT). While no QRT is available for general non-Markovian open quantum systems, we show that for time-independent Hamiltonians and finite memory times

τ_c

, an exact factorization rule exists that relates higher-order multitime correlations to products of lower-order correlations. Consequently, all information needed to reconstruct

n

-time correlations is contained in a temporal volume of

\mathcal{O}(τ_c^n)

. On the example of quantum dots coupled to phonons, we demonstrate that this factorization makes numerical calculations of multitime correlations extremely efficient and even enables semianalytical solutions in systems where the standard QRT breaks down.

The quantum regression theorem is one of the central results in open quantum systems and is extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schr\"odinger picture. In this paper we make use of the Heisenberg picture to derive the quantum regression theorems for multi-time correlation functions, which in the special limit reduce to the well-known two-time regression theorem. For the multi-time correlation function we find that the regression theorem takes the same form as it takes for the two-time correlation function with a mild restriction that one of the times should be greater than all other time variables. Interestingly, the Heisenberg picture also allows us to derive an analog of regression theorem for out-of-time-ordered correlators. We further extend our study for the case of non-Markovian dynamics and report the modifications to the standard quantum regression theorem. We illustrate all of the above results using the paradigmatic dissipative spin-boson model.

We investigate the presence of memory in the sequential measurement statistics of an open quantum system, as witnessed by the departure from the quantum regression theorem (QRT), that is, the possibility to predict multitime probabilities from the one-time reduced dynamical map. For factorized initial states, we identify an exact decomposition of the two-time propagator into a QRT-like contribution, fully determined by the reduced dynamical map, and a memory term encoding system--environment correlations across the intervention; in the weak-coupling regime, the memory term yields an explicit second-order correction expressed in terms of the reduced map and bath correlation functions. Furthermore, we introduce an operational quantifier of QRT violations based on the distance between exact and QRT-predicted joint probabilities. Benchmarking the framework on a spin--boson model and using a pseudomode embedding as nonperturbative reference, we comprehensively analyze the impact of spectral-density parameters, environmental temperature, and measurement protocols on the non-Markovianity of the multitime statistics. Comparison with a one-time quantifier shows that reduced-state non-Markovianity and multitime memory are related but inequivalent: the latter, as probed through sequential statistics, is intrinsically protocol dependent and can become visible at higher temporal order even when two-time statistics remain compatible with QRT predictions.