臺灣博碩士論文加值系統

在更高維度的純態雙分系統裡我們展示了糾纏的精細描述。除了二乘二系
統以外，我們需要一個以上的參數來描述糾纏。一個系統非糾纏性的充分必要
條件是所有的糾纏參數都為零的狀況，每個糾纏參數都可視為一個二乘二系統
的糾纏。這邊將探討這些糾纏參數的各種性質，而且重點放在二乘三的系統，
此種系統是需要多於一個糾纏參數的最簡單的系統。其中探討了糾纏參數的對
稱性、可交換與非交換Hopf纖維化和糾纏參數的精確對應、糾纏參數的測
量，最後提供明確的物理系統來展示這些糾纏參數的表現行為。

Finer characterizations of entanglement have been shown to exist in higher dimensional pure bipartite systems. Beyond qubit-qubit systems, more than one parameter is necessary to capture the entanglement. The necessary and suffi cient condition for separability is equivalent to the vanishing of a set of entanglement parameters, each of which can be associated with a qubit-qubit entanglement. Properties of these entanglement parameters are investigated, and attention is focussed on the qubit-qutrit system-the simplest system which requires more than one parameter of entanglement.Symmetries in the entanglement parameters, and the precise parametrization of entanglement in Abelian and non-Abelian Hopf brations of the Hilbert space are also
discussed. Measurement of the entanglement parameters is also addressed, and explicit physical examples are used to verify and exemplify the behavior of the entanglement parameters.

Contents
1 Introduction and overview ... 3
2 A brief review of ner entanglement parameters for pure bipartite systems ... 4
3 Symmetries of entanglements ... 5
3.1 Entanglements symmetry on 2 by m system ... 5
3.2 Symmetry on total entanglement of n by m state ... 6
4 Finer characterizations of entanglement parameters: Explicit examples ... 10
4.1 6-state Hopf bration and entanglement ... 10
4.2 Abelian Hopfbration ... 11
4.3 Non-abelian quaternionic Hopf bration ... 12
5 Measurement of entanglement parameters ... 15
6 Entanglement of spin-spin interaction: 4 and 6-state examples ... 20
7 Example of 6-state decomposition from total spin state to bipartite qubit-qutrit system ... 24
7.1 Rotating magnetic eld couple with two spin ... 24
7.2 Clebsch-Gordan division into bipartite state ... 28
7.3 Calculation on entanglement parameters ... 30
7.4 Time dependence of the entanglement parameters ... 31
8 Concluding remarks ... 38
References ... 39

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