透過四元數Hopf纖維化S^(4l+3)/S^3≅〖"HP" 〗^l，我們研究了任意偶數維SU(2)量子幾何相。量子態重疊函數的Schrӧdinger演化透過用基空間和纖維的座標的表達顯現出SU(2)幾何相〖"T" e〗^(-i∫_0^T▒Adt)和動力相"T" e^(i/ℏ ∫_0^T▒〖"H" dt〗)。所得的公式是複數Hopf纖維化S^(2N+1)/S^1≅〖"CP" 〗^N的U(1)幾何相在四元數Hopf纖維化的SU(2)類比。這些成果適用於任意Hamiltonian和任意偶數維的量子態，也是在沒有作絕熱近似條件下對於開放或封閉演化路徑都有效。我們用S^7/S^3≅〖"HP" 〗^1作為一個例子在三種不同看法下得到: (1)BPST 瞬間子的幾何基礎S^7/SU(2)≅S^4，(2)交換的複數Hopf纖維化 S^7/U(1)≅〖"CP" 〗^3，以及(3)一個雙量子位元系統的糾纏態空間為S^7[S^3xS^3]。我們討論了這些看法之間的關係以及瞬間子和糾纏參數兩者的精確對應，也討論了廣義的Wilczek-Zee幾何相，且示範了四元數SU(2)幾何聯絡及其和Wilczek-Zee-Berry聯絡的關係。

Exact non-abelian geometric phase for arbitrary even-dimensional quantum systems is investigated through quaternionic Hopf fibrations S^(4l+3)/S^3≅HP^l. Time evolution according to Schrӧdinger equation of the generic state is expressed in terms of quaternionic base manifold and fiber coordinates, and the non-abelian phase factor, Te^(-i∫_0^T A dt), as well as the quaternionic analog of the dynamical phase factor, Te^(i/ℏ ∫_0^T H dt), are manifested in the overlap function. The formula derived is the non-abelian quaternionic Hopf fibration S^(4l+3)/S^3≅HP^l analog of the abelian geometric phase construction of the complex Hopf fibration S^(2N+1)/S^1≅CP^N. The result obtained holds for arbitrary even-dimensional systems which obey Schrӧdinger evolution with arbitrary Hamiltonian, and is valid without adiabatic approximation for both closed and also open paths. S^7/S^3≅HP^1 is discussed as an explicit example viewed from three different perspectives: (1) S^7/SU(2)≅S^4 as the geometrical basis of the Belavin-Polyakov-Schwartz-Tyupkin instanton, (2) S^7 as the Hilbert space of a 4-state system for the abelian complex Hopf fibration S^7/U(1)≅CP^3, and (3) as the Hilbert space of a bipartite qubit-qubit system with the space of entangled states identified with S^7[S^3×S^3]. Explicit relations between these constructions are discussed and the precise correspondence between instanton parameter and entanglement is also clarified. The derivation of non-abelian Wilczek-Zee phase is carried out in a general context and explicit illustrations of the exact non-abelian SU(2) geometric connection and its relation to the Wilczek-Zee-Berry connection is provided.

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