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arXiv:2510.19987 · Physical Review A

Separation of quantum time evolution into holonomic and dynamical parts

Adam Fredriksson, Erik Sjöqvist

Published May 13, 2026• 0 citations
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Abstract

The issue of separating Schrödinger-type quantum time evolution into a product of holonomic and dynamical parts in the nonadiabatic non-Abelian case is examined. We identify all special cases in which this kind of separation is possible, and we prove that separability is a gauge-invariant property of quantum time evolution. The general analysis is implemented in a three-level system with Λ -type coupling structures. The typical case where the holonomic and dynamical parts do not separate is illustrated by means of two noncommuting Λ -type Hamiltonians.

DOI
10.1103/n9tc-4ppk

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Comment on "Evolution Operator Can Always Be Separated into the Product of Holonomy and Dynamic Operators"

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We show that the claim in Ref. [PRL 131, 200202 (2023)], that the quantum time evolution always can be written as a product of a holonomy operator and a dynamic operator, is false, as it is based on a circular use of the time evolution operator.

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The issue of separating Schrödinger-type quantum time evolution into a product of holonomic and dynamical parts in the nonadiabatic non-Abelian case is examined. We identify all special cases in which this kind of separation is possible, and we prove that separability is a gauge-invariant property of quantum time evolution. The general analysis is implemented in a three-level system with Λ -type coupling structures. The typical case where the holonomic and dynamical parts do not separate is illustrated by means of two noncommuting Λ -type Hamiltonians.
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.
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.
We investigate whether a single-loop construction in non-adiabatic holonomic quantum computation can be generalized to multi-segment evolutions while preserving separation between geometric and dynamical contributions. Starting from a three-level Λ-system, we first study phase-only basis transformations between segments and find an exact non-trivial three-segment solution under correlated pulse-area and phase constraints. We then use a numerical loss-function approach to search for zero-leakage solutions with more than three segments, and find evidence of irreducible multi-segment evolutions in the tested low-segment cases. We further investigate more general unitary basis transformations and analyze the separation condition using projected dynamical generators in the computational subspace. The analysis suggests that admissible transformations must preserve a common dark direction, effectively reducing the problem to a two-dimensional bright-excited subspace. While an enlargement of the set of reachable single-qubit gates is not possible, the results may provide additional implementation freedom for realizing holonomic gates.
Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.
We discuss separability of solutions to a Schrödinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schrödinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product P A ⊗ P B of two projections on the subsystems K A and K B , respectively, then the state |ψ(t)〉 of the composite system starting from a separable initial |ψ(0)〉 = |ψ A 〉 ⊗ |ψ B 〉 is separable for all t ∈ [0, T] if and only if either |ψ A 〉 is an eigenstate of P A , or |ψ B 〉 is an eigenstate of P B .