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.
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.
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.
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.
The quantum adiabatic theorem conventionally posits that a quantum system initialized in an instantaneous eigenstate of a time-dependent Hamiltonian will remain in that corresponding instantaneous eigenstate, provided the parameter changes are sufficiently slow and no level crossings occur. Under this idealized framework, standard quantum mechanics neglects any intrinsic energetic cost associated merely with traversing a trajectory through the projective Hilbert space or the underlying parameter manifold, assuming that adiabatic evolution is inherently frictionless and dissipation-free in closed systems. In this work, we challenge this traditional paradigm by introducing the concept of Quantum-State Path Friction (QSPF). We propose that even under strictly ideal adiabatic conditions, an irreducible energetic penalty exists, which scales quadratically with the velocity of the quantum state along its geometric path. We formally define the path work as W_path = \int \eta_Q (ds_Q / dt)^2 dt, where ds_Q^2 = g_{ij} d\lambda_i d\lambda_j denotes the quantum geometric line element governed by the quantum metric tensor g_{ij}, and \eta_Q represents the quantum friction coefficient. This formulation establishes a foundational framework for a new class of dissipation laws in quantum dynamics, bridging quantum information geometry, non-equilibrium thermodynamics, and optimal quantum control.
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.
Quantum mechanics has both deepened our understanding of the physical reality and allowed development of quantum technologies beating their classical counterparts in efficiency, security, and sensitivity. As a major downside, the quantum properties behind these technologies are very sensitive to the effects the system’s interaction with its environment. The theory of open quantum systems describes how such interaction influences the system, and as a consequence, how the quantum features are lost, controlled, and possibly returned by manipulating the environment. This thesis begins with a short description of the most relevant features and results of the open quantum system theory. We continue by presenting some commonly used measures to quantify the information encoded in quantum systems and the correlations between spatially or temporally separated parties. The literature review ends with discussion of non-Markovianity and memory effects which combine the fields of open quantum systems and quantum information in a natural way. The rest of the thesis presents the main results of Publications I – VII. We study the memory effects and physicality conditions of open system dynamics described by the vast family of phase-covariant master equations. We show how the open quantum system picture can be exploited to design probing protocols to extract information without knowing how the probe and system interact. Such protocols are considered also from the point-of view of efficiently obtaining the information required in the probing, both in theory and by experimental implementation. The experiment serves also as evidence for applicability of recently developed geometric tools to design informationally incomplete measurements for solving so-called membership problems. We present and experimentally realize a protocol where control over intial correlations between polarization and frequency gives full freedom to implement any dephasing dynamics in the open system polarization, thus dictating how its information content changes in time. We study the two-qubit memory effects of dephasing dynamics and compare them to the effect of such noise in the transmission efficiency in superdense coding. Optimal noise configurations in different situations are found. Finally, we characterize the families of initial states for which the decay of quantum correlations depends or does not depend on the noise location.
A fundamental challenge of quantum computation is being able to scale up a large number of high fidelity quantum gates while noise and error are affecting the gate’s physical control parameters. This dissertation focuses on the fidelity of single-qubit quantum gates constructed by a change in quantum geometric phase, while the control parameters are affected by random noise and systematic error. A unified model of geometric quantum computation is developed, in which a qubit state is controlled by a composite Hamiltonian, resulting in a multiple-segment rotation of the quantum state and allowing characterization of evolution paths depending on the associated geometric and dynamic phase. The fidelity of the quantum gates in the presence of different noise error is compared for pu rely geometric, hybrid (having both geometric and dynamic phase), and conventional dynamic quantum gates built on single Hamiltonians. Results showed hybrid quantum gates had the highest fidelities, followed by geometric gates, and conventional dynamic gates had the lowest fidelities. In addition, there was indication in some cases higher fidelities result from gates created from a larger number of segments in the quantum state rotation. These results can be understood by the relation of the control parameters with the evolution path geometry. By translating between control parameters, our model can be applied to different systems for quantum computation, including: the laser manipulation of a two-level atom, laser manipulation of trapped ions, nuclear magnetic resonance, polarization states of photons, superconducting qubits in cavity QED, and quantum dots.