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arXiv:2607.05218 · Journal of Physics A Mathematical and Theoretical

Quasi-holonomy in non-adiabatic quantum evolution

Erik Sjöqvist, Adam Fredriksson

quant-phSubmitted Jul 6, 2026• Announced Jul 7, 2026• 0 citations

v1 · Jul 6, 2026 v2 · Sep 3, 2026

Abstract

We develop a framework for quasi-holonomy in non-adiabatic quantum time evolution of subspaces along loops in a complex Grassmannian. By factoring the Schrödinger evolution into dynamical and connection-induced contributions in a moving basis, we obtain an effective geometric generator that depends explicitly on the dynamical propagator. This quasi-connection does not define a genuine connection on the original Grassmann bundle, since its gauge transformation law acquires a history-dependent, nonlocal term. Other ways of factoring the Schrödinger evolution are briefly discussed. All these approaches suffer from the same type of history-dependence, thereby defining transport of subspaces in which geometric and dynamical effects are generally intertwined, just as in the case of the quasi-holonomy. Our work sheds light on the issue of separating quantum evolution of subspaces into holonomic and dynamical parts from an essentially gauge-theoretic perspective.

Journal reference: J. Phys. A: Math. Theor. 59, 355308 (2026)
DOI: 10.1088/1751-8121/ae9e6e

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The geometric phase is a fundamental quantity characterizing the holonomic feature of quantum systems. It is well known that the evolution operator of a quantum system undergoing a cyclic evolution can be simply written as the product of holonomic and dynamical components for the three special cases concerning the Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase. However, for the most general case concerning the nonadiabatic non-Abelian geometric phase, how to separate the evolution operator into holonomic and dynamical components is a long-standing open problem. In this Letter, we solve this open problem. We show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, we further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. Our finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.

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The proposal of the optical scheme for holonomic quantum computation is evaluated based on dynamical resolution to the system beyond adiabatic limitation. The time-dependent Schrödinger equation is exactly solved by virtue of the cranking representation and gauge transformation approach. Besides providing rigorous confirmation to holonomies of the geometrical prediction that holds for the ideally adiabatic situation, the dynamical resolution enables one to evaluate elaborately the amplitude of the nonadiabatic deviation, so that the errors induced to the quantum computation can be explicitly estimated.

When a quantum state evolves in such a way as to describe a closed loop in the space of pure state density matrices, it must, as a consequence of the non trivial topology of this space acquire a path-dependent phase. When the state vector | ψ 〉 evolution is such that < ψ | d/dt | ψ > = 0, the resulting phase is that introduced by Aharanov and Anandan (thereafter called the A.A. phase). Mathematically this condition corresponds to a parallel transport of | ψ 〉 by a connection defined on a fiber bundle. This paper contains an elementary and self-contained discussion of the A.A. phase for a spin-1 system. In this case, the phase appears as the holonomy of the natural connection over the complex projective space P 2(C). Experimental verification of these ideas requires expressions for both the phase in terms of the path and a Hamiltonian which will parallely transports the state vector along the path. They are given in terms of four directly measurable quantities which parametrize the pure state spin-1 density matrices. It is not possible to measure directly the A.A. phase on an isolated system ; it requires the separating and subsequent bringing together of two subsystems which undergo different evolutions. We suggest two ways in which, in principle, the A.A. phase might be measured in the laboratory.

Abstract We introduce a phase-geometric framework in which time evolution is governed by a covariant derivative defined over a multi-carrier phase space. The associated connection induces a non-abelian curvature whose holonomy generates measurable phase-transport effects. Within this unified geometric setting, the quantum–classical transition, phase-channel mixing, and time-transport anomalies arise from a single underlying mechanism. In contrast to conventional approaches, where gauge structures are imposed externally, the connection emerges intrinsically from the phase geometry itself. The framework yields calculable thresholds for classicalization and leads to novel, testable predictions based on curvature-induced phase holonomy.