本篇文章探討量子態的數學結構，首先定義純態與混合態，並以密度矩陣的形式描述量子態的集合結構。對於複合系統，文中引介雙系統（bipartite system）之概念，利用偏跡運算解析部分系統之狀態，並進一步說明混合態可由純態的凸組合表示。
第三章專注於量子糾纏與可分離性準則。對於純態，文中證明Schmidt 階數為判別可分離性的重要依據。混合態部分，則介紹正映射準則（Positive Map Criterion）、部分轉置準則（Positive Partial Transpose Criterion）與歸約準則（Reduction Criterion）等工具。這些準則提供可實際操作之方法，以辨識混合態是否為糾纏態。
第四章提出Entanglement witness 理論。該理論利用希爾伯特空間上之Hermitian 算符構造，可有效區分糾纏態與可分離態。此方法在實務上尤為關鍵，因其能透過實驗觀測間接驗證糾纏存在。透過上述內容，本文建立量子糾纏判別理論之基礎，為後續量子資訊處理與通道理論研究提供重要工具。

This thesis investigates the mathematical structure of quantum states, with emphasis on separability and entanglement detection. Chapter 2 outlines the definitions of pure and mixed states using density matrices, and introduces the notion of bipartite systems. Reduced states are obtained via partial trace, and the convex structure of the state space is highlighted.
Chapter 3 presents key criteria for separability. For pure states, the Schmidt decomposition provides a necessary and sufficient condition: a state is separable if and only if it has Schmidt rank one. For mixed states, the Positive Map Criterion, the Positive Partial Transpose (PPT) Criterion,
and the Reduction Criterion are introduced as tools to detect entanglement, though their effectiveness is dimension-dependent.
Chapter 4 introduces entanglement witnesses—Hermitian operators that detect entangled states by yielding negative expectation values on them while remaining non-negative on all separable states. This operational approach is particularly valuable in experimental settings. Together, these chapters provide a compact yet rigorous foundation for analyzing quantum correlations, with practical implications in quantum information theory.