量子偵測問題的目的是使用量子力學來設計最佳化接收器，以求達到最小的偵測錯誤機率。如果每個被傳送的訊號可以用量子純態向量來表示，我們可以把訊號偵測歸類為兩種情況。當每個訊號所對應的向量是相互正交的，我們可以使用凡紐曼(von Neumann)量測來實現一個無錯誤的通訊系統。相反地，如果訊號所對應的向量是非相互正交的，則沒有一種接收器，或稱為量子量測運算子，可以完美地偵測這些訊號，這就是量子偵測理論本質上所探討的問題。因此，我們討論的是非相互正交的訊號。對於這些訊號，我們希望能設計出最佳化量子量測運算子。

本篇論文著重在討論三位量子純態的偵測問題。首先，我們研究了在艾爾達(Eldar)論文裡面所提到關於量子訊號偵測問題的相關理論。我們根據她在同機率之幾何均勻(Geometrically Uniform)量子純態的研究結果，使用矩陣形式來表示同機率之三位幾何均勻量子純態的最佳化量測運算子。接下來，我們考慮不同機率之三位幾何均勻量子純態，在此情況下，我們假設其中一個量子純態的優先機率已知，其餘則否。為了解決這個問題，我們建立了最佳化問題，推導出最佳化解答的充分且必要條件，並且使用數值模擬的方法，以求出最佳化量測運算子的解析解。根據數值模擬的結果，我們對最佳化優先機率以及最佳化量測運算子的解析解做了假設。

This thesis concerns the quantum detection problem for ternary pure state signals. We first review the relevant study on the quantum signal detection problem in the manner determined in Eldar's paper. We verify her result with the ternary geometrically uniform (GU) pure state signal having the uniform prior probability distribution, in which we have used the matrix-form expression of the optimum detection operators. Next, we consider the optimum detection problem for the ternary GU pure state signal
under the condition that one prior probability is given and the others are unknown. In this situation, we formulate our
optimization problem and derive the necessary and sufficient
conditions for the solution of the problem. To seek the
closed-form expression for the optimum detection operators, we perform the numerical simulation. As a result, we make a
conjecture on the closed-form expression of the optimum measurement operators and on the optimum prior distribution for our problem.

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