  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 work, 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.
