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.
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.
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 τc, 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 O(τcn). 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.
In the present Thesis we study the behavior of multi-time correlation functions and of thermodynamical quantities such as heat in open quantum systems undergoing an evolution generally affected by the presence of memory effects, i.e. non-Markovian. In the last decade, a large part of the scientific community in this field has dedicated its efforts to the understanding, precise definition and quantification of non-Markovianity in the quantum realm and now we have at our disposal several benchmark results and a plethora of different estimators that allow to determine the degree of non-Markovianity of a given dynamics. It comes therefore natural to investigate how other different dynamical quantities relate to such estimators also in order to understand the physical implications of memory effects on the statistics of observable quantities. In the first part of this work, a quantitative test of the violation of the so-called quantum regression theorem in presence of a non-Markovian dynamical regime is investigated. The quantum regression theorem represents a procedure that, whenever valid, allows to reconstruct two-time correlation functions of system's operators from the sole knowledge of the dynamics of mean values. It is worth stressing that two-time correlation functions are necessary in order to fully characterize the statistical properties of a quantum system, since they are able to catch aspects of the dynamics, such as fluorescence spectrum, in general not accessible looking at mean values. Despite their relevance however, obtaining two-time correlation functions often represents a formidable task, since the knowledge of the full "system+environment" dynamics is required, a generally too demanding request in the context of open quantum systems theory. The quantum regression theorem represents in this regard the easiest route to determine two-time correlation functions, this highlighting its importance. In this work we show that, in a pure-dephasing spin-boson model, the quantum regression theorem represents a stronger condition than non-Markovianity, in the sense that any presence of memory effects in the reduced dynamics inevitably results in violations to the former. These results have been published in [G.Guarnieri, A. Smirne, B. Vacchini, Phys. Rev. A 90, 022110 (2014)]. The second part of the Thesis is devoted to the characterization of heat ow at the microscopic level in open quantum systems, both finite and infinite dimensional. In particular we begin by studying the time behavior of its mean value in a non-Markovian dynamical regime, showing that, at variance with what happens in the Born-Markov semigroup limiting case, heat can backflow from the environment to the system. After providing a condition for the occurrence of such phenomenon and a measure for its amount for a given dynamics, the relationship with suitable non-Markovianity estimators is sought in two paradigmatic models, namely the spin-boson and the quantum brownian motion. The results, collected in [ G. Guarnieri, C. Uchiyama, B. Vacchini, Phys. Rev. A 93, 012118 (2016); G. Guarnieri, J. Nokkala, R. Schmidt, S. Maniscalco, B. Vacchini, Phys. Rev. A 94, 062101 (2016)], on the one hand allow for the identification of parameter-regions where the heat backflow is absent or maximum. On the other hand they show that the occurrence of heat backflow represents a stricter condition than non-Markovianity, in the sense that non-Markovianity allows for the observation of heat flowing back from the environment to the system and, vice versa, a Markovian dynamics prevents its occurrence. This Thesis concludes with the formulation of a new family of lower bounds to the mean dissipated heat in an environmental-assisted erasure-protocol scenario where Landauer's principle applies. As originally conceived for classical systems, this principle states that every irreversible erasure of information stored in a system inevitably carries along an amount of heat dissipated into the environment which is expended to perform the action. Within the framework recently put forward in [D. Reeb, M. M. Wolf, New J. Phys. 16, 103011 (2014)], which guarantees the validity of Landauer's principle in an open quantum systems scenario, we provide an asymptotically tight family of lower bounds to the dissipated heat which are also valid in the non-equilibrium setting. This construction is applied to an open system consisting of a three-level V-system, in which one transition is externally pumped by a laser field while the other is coupled through an XX-interaction to an environment consisting of a spin chain. Beside calculating all these quantities, an exact solution for the dynamics of such system is also provided. These results are collected in [G. Guarnieri, S. Campbell, J. Goold, S. Pigeon, M. Paternostro, B. Vacchini, in preparation].
We explore the connection between two recently introduced notions of non-Markovian quantum dynamics and the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamics has been defined looking at the behavior in time of the statistical operator, which determines the evolution of mean values, the quantum regression theorem makes statements about the behavior of system correlation functions of order two and higher. The comparison relies on an estimate of the validity of the quantum regression hypothesis, which can be obtained exactly evaluating two-point correlation functions. To this aim we consider a qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of the non-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities. We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itself violation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.
We investigate the nature of memory effects in the non-Markovian dynamics of spin-boson models. Local quantum memory criteria can be used to indicate that the reduced dynamics of an open system necessarily requires a quantum memory in its environment. We focus on two such criteria, one based on the dynamical map and the other on single-intervention process tensors, and discuss their connection. We apply these criteria to quench dynamics in spin-boson and two-spin-boson models, computed using a numerically exact method for non-Markovian open-system dynamics based on matrix product operator influence functionals. This method is ideally suited to simulate quantum multitime processes in spin-boson models across broad parameter regimes. We find that, with access to single-intervention process tensors, one can generally predict quantum memory in the dynamics at low temperatures. Given instead only the dynamical map, it is still possible to detect quantum memory in the case of resonant environments at short evolution times. Moreover, we confirm quantum memory in the stationary dynamical regime using process tensors with the correlated steady state of system and environment as initial condition.
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.
While the quantum regression theorem (QRT) is the standard tool for computing multi-time correlation functions in open quantum systems, it relies on system-bath separability and an environment that remains in equilibrium, assumptions that are violated once dynamical correlations develop. Using the projection operator formalism, we derive an extension to the QRT that explicitly incorporates these correlation-induced corrections, enabling us to capture multi-time correlation functions in non-Markovian regimes. We apply this framework to the variational polaron master equation for the spin-boson model in ohmic and super-ohmic regimes, where the polaron transformation mixes system-bath degrees of freedom to produce a non-thermal effective environment. Benchmarking against numerically exact tensor-network simulations, we demonstrate quantitative agreement for single- and two-time observables, including linear-response spectra, even at strong coupling. Our approach broadens the reach of analytic master equations to multi-time observables in strong-coupling and non-Markovian regimes, where environmental memory effects and system-bath correlations are crucial.