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Probing Non-Markovianity through Topological Signatures of Path Space

PAULO SÉRGIO DE ANDRADE

Published Nov 24, 2025• 0 citations
Publisher

Abstract

The characterization of non-Markovian dynamics in open quantum systems remains a central challenge in quantum physics. Conventional measures often rely on specific properties of the system's evolution, such as the flow of information or the divisibility of the dynamical map. This paper introduces a fundamentally different approach by connecting quantum memory effects to the topological structure of the system's path space. We posit that the set of all possible quantum trajectories, when viewed as a geometric object, exhibits distinct topological signatures under non-Markovian evolution compared to its Markovian counterpart. Using tools from topological data analysis, specifically persistent homology, we analyze the path space generated from simulations of a canonical open quantum system. We demonstrate that topological invariants, such as the Betti numbers, which count holes and voids in the path space, serve as robust indicators of non-Markovianity. Our results reveal that the presence of quantum memory induces non-trivial topological features, such as persistent loops, in the space of paths. This suggests that memory effects constrain the system's evolution, forcing trajectories to organize into complex, non-contractible structures. This topological framework provides a model-agnostic and potentially more robust method for detecting and quantifying quantum memory, opening new avenues for understanding complex quantum dynamics from a geometric perspective.

DOI
10.5281/zenodo.17699510

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The characterization of non-Markovian dynamics in open quantum systems remains a central challenge in quantum physics. Conventional measures often rely on specific properties of the system's evolution, such as the flow of information or the divisibility of the dynamical map. This paper introduces a fundamentally different approach by connecting quantum memory effects to the topological structure of the system's path space. We posit that the set of all possible quantum trajectories, when viewed as a geometric object, exhibits distinct topological signatures under non-Markovian evolution compared to its Markovian counterpart. Using tools from topological data analysis, specifically persistent homology, we analyze the path space generated from simulations of a canonical open quantum system. We demonstrate that topological invariants, such as the Betti numbers, which count holes and voids in the path space, serve as robust indicators of non-Markovianity. Our results reveal that the presence of quantum memory induces non-trivial topological features, such as persistent loops, in the space of paths. This suggests that memory effects constrain the system's evolution, forcing trajectories to organize into complex, non-contractible structures. This topological framework provides a model-agnostic and potentially more robust method for detecting and quantifying quantum memory, opening new avenues for understanding complex quantum dynamics from a geometric perspective.
ABSTRACT: We use the divisibility approach to describe non-Markovian dynamics (memory effects) in open quantum systems coupled to an environment (reservoir). Specifically, sufficient and necessary criteria were incorporated in the energy domain that play a relevant role in dynamics of the system. In this case, we adopt the Feshbach partition in the space of (Hilbert-Liouville). On the other hand, to investigate both Markovianity and non-Markovianity, we study the divisibility of the Wigner function propagator. For this, within the framework of path integrals, we carry out an analysis of non-classical paths in phase space. Furthermore, we have used tools of topology to propose theorems intimately related to measures of non-Markovianity that are experementally accessible.
The reduced description of the quantum dynamic processes in the condensed phase environment leads to the equation of motion with a memory kernel. Such a memory effect, termed non-Markovianity, presents more complex dynamics compared to its memoryless or Markovian counterpart, and many chemical systems have been demonstrated through numerical simulations to exhibit non-Markovian quantum dynamics. Explicitly how the memory impacts the dynamic process remains largely unexplored. In this work, we focus on ways to separate the non-Markovian contributions from the dynamics and study the non-Markovian effects. Specifically, we developed a rigorous procedure for mapping the exact non-Markovian quantum propagator to the Lindblad form. Consequently, it allows us to extract the negative decay rate from the Lindbladian that is the signature of the non-Markovianity. By including or excluding the negative rate in the time evolution, we can decisively pinpoint the influence of non-Markovianity on the system's properties such as coherence, entanglement, and equilibrium state distribution. The understanding of such memory effects on the dynamic process suggests the possibility of leveraging non-Markovianity for quantum control.
Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T({t}_{2},{t}_{0})\ensuremath{\ne}T({t}_{2},{t}_{1})T({t}_{1},{t}_{0})$ in terms of memoryless dynamical map TTT introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
This Thesis discusses the phenomenology of the dynamics of open quantum systems marked by non-Markovian memory effects. Non-Markovian open quantum systems are the focal point of a flurry of recent research aiming to answer, e.g., the following questions: What is the characteristic trait of non-Markovian dynamical processes that discriminates it from forgetful Markovian dynamics? What is the microscopic origin of memory in quantum dynamics, and how can it be controlled? Does the existence of memory effects open new avenues and enable accomplishments that cannot be achieved with Markovian processes? These questions are addressed in the publications forming the core of this Thesis with case studies of both prototypical and more exotic models of open quantum systems. In the first part of the Thesis several ways of characterizing and quantifying non-Markovian phenomena are introduced. Their differences are then explored using a driven, dissipative qubit model. The second part of the Thesis focuses on the dynamics of a purely dephasing qubit model, which is used to unveil the origin of non-Markovianity for a wide class of dynamical models. The emergence of memory is shown to be strongly intertwined with the structure of the spectral density function, as further demonstrated in a physical realization of the dephasing model using ultracold quantum gases. Finally, as an application of memory effects, it is shown that non- Markovian dynamical processes facilitate a novel phenomenon of timeinvariant discord, where the total quantum correlations of a system are frozen to their initial value. Non-Markovianity can also be exploited in the detection of phase transitions using quantum information probes, as shown using the physically interesting models of the Ising chain in a transverse field and a Coulomb chain undergoing a structural phase transition.
The conventional understanding of quantum phase transitions (QPTs) focuses on the initial and final states of a system undergoing a change in a control parameter, often neglecting the trajectory taken between those endpoints. In this work we introduce the concept of Quantum Phase Path Memory (QPPM), a theoretical framework that attributes the final quantum state not only to the boundary conditions but also to the specific path Γ(t) followed through the parameter space. By extending the adiabatic theorem to include path-dependent corrections and by analyzing the role of geometric phases such as the Berry phase, we demonstrate that the memory of the evolution path is encoded in the phase evolution of the system's wavefunction. We develop a formalism that quantifies this memory effect using a path integral representation of the unitary propagator and show that the accumulated phase can be decomposed into an intrinsic dynamical part and a geometric part that depends explicitly on the trajectory in parameter space. The implications of QPPM are wide-ranging: it provides a new lens for interpreting critical phenomena, offers novel strategies for controlling quantum states in adiabatic quantum computation, and suggests experimental signatures in systems where the control parameter can be modulated in time. While no experimental data are presented here, the theoretical predictions point toward feasible tests in ultracold atoms, superconducting qubits, and topological insulators, where the control of the Hamiltonian is highly precise. Finally, we discuss potential extensions of QPPM to non-Hermitian dynamics and to open quantum systems, and we outline how the memory effect could influence the design of quantum simulators and error correction protocols.