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Quantum-State Path Friction (QSPF): A Fundamental Dissipative Law in Adiabatic Quantum Dynamics

Jincheng Zhang

Published Aug 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21888335

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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.
Traditional quantum mechanics and thermodynamics predominantly treat energy as a state function, focusing on initial and final states while treating the intermediate trajectory as a secondary detail dictated solely by the equations of motion. In this paper, we challenge this paradigm by introducing the framework of Quantum Energy-Path Curvature (QEPC). We propose that between the exact same initial state and final state, different quantum evolution paths can incur fundamentally distinct effective energy costs. By defining a rigorous path energy functional and a corresponding path curvature metric over the control and state manifold, QEPC reveals a deeper geometric structure underlying quantum transitions. We formulate the mathematical foundations of path energy and curvature, derive key properties governing quantum evolution under geometric constraints, and present testable predictions where two distinct paths connecting identical initial and final states yield differing energy expenditures (E_Gamma_1 != E_Gamma_2). This framework broadens the conceptual boundaries of energy state functions and offers profound implications for quantum control, quantum thermodynamics, and fundamental quantum mechanics.
Transitionless quantum driving is a useful approach to mimic the adiabatic evolution of a quantum system so that the system can rapidly evolve from its initial state to the target state without transitions between instantaneous eigenstates of the reference Hamiltonian. To quantify the energetic cost in the transitionless quantum driving process, the instantaneous cost is defined as the Frobenius norm of the driving Hamiltonian and the integral of instantaneous cost is used to describe the total cost in the entire evolution. In this paper, we find that the minimal integral of instantaneous cost has a geometric property being equal to the length of the evolution path on the Riemannian manifold spanned by the control parameters with a positive definite metric, but independent of the evolution details such as the changing rate of the parameters. Based on this property, we further show that the optimal transitionless quantum driving with the minimum total cost corresponds to the geodesic path on the Riemannian manifold, which provides a method for optimizing transitionless quantum driving.
Keeping a quantum system in a given instantaneous eigenstate is a control problem with numerous applications, e.g., in quantum information processing. The problem is even more challenging in the setting of open quantum systems, where environment-mediated transitions introduce additional decoherence channels. Adiabatic passage is a well-established solution but requires a sufficiently slow evolution time that is dictated by the adiabatic theorem. Here we develop a systematic projection theory formulation for the transitionless evolution of general open quantum systems described by time-local master equations. We derive a time-convolutionless dynamical equation for the target instantaneous eigenstate of a given time-dependent Hamiltonian. A transitionless dynamics then arises in terms of a competition between the average Hamiltonian gap and the decoherence rate, which implies optimal adiabaticity timescales. We show how eigenstate tracking can be accomplished via control pulses, without explicitly incorporating counter-diabatic driving, thus offering an alternative route to accelerate adiabaticity. We examine rectangular pulses, chaotic signals, and white noise, and find that, remarkably, the effectiveness of eigenstate tracking hardly depends on the details of the control functions. In all cases the control protocol keeps the system in the desired instantaneous eigenstate throughout the entire evolution, along an accelerated adiabatic path.
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.
Quantum adiabaticity is the evolution of a quantum system that remains close to an instantaneous eigenstate of a time-dependent Hamiltonian. Using Floquet formalism, we derive a rigorous sufficient condition for adiabaticity in closed, finite-dimensional periodically driven systems that is valid for arbitrarily many driving periods. The condition is stroboscopic and geometric, depending only on single-cycle information: the Fubini--Study length of the instantaneous eigenray and a quasienergy-separation measure extracted from the Floquet operator. We also formulate a state-targeted refinement that reduces conservativeness when only one adiabatic branch is relevant. Rather than synthesizing control pulses, the result provides a certification criterion for a given periodic protocol. We illustrate the criterion and contrast it with conventional instantaneous-gap conditions in three representative examples.