在研究量子資訊的領域中，對於如何去分類、描述各種型式的量子態（quantum state），進一步地去了解何謂量子纏結（quantum entanglement），是近幾年來相當重要的課題。就數學問題的角度來看，我們必須去探討任意一個多重量子態（multipartite quantum states）系統在局域酉轉換（local unitary transformation）之下，如何藉由在這轉換之下的不變量描述這系統所表現出來的特性。因此我們在這篇論文中，提供一套明確的方法---combinatorial tracing，幫助我們找出所有所需要的不變量，並且探討這些不變量之間的關係，而這整個過程可以視為廣義的Schmidt分解。

To characterize different types of quantum states is one of major inquiries in the area of quantum information. In this thesis a general scheme, combinatorial tracing, is developed, that enables the calculation of all the essential algebraic invariants for a quantum state. These invariants are considered registers or indexes that characterize an arbitrary, pure or mixed, multipartite quantum state. Specifically they provide the necessary and sufficient condition to determine whether two arbitrarily given states are equivalent up to local unitary transformations. This scheme is in practice a generalized
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