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Notation

Dirac Notation

就是另一種表示 vector 的方式,這裡的數字都是複數

|\mathbf{v}\rangle = \begin{bmatrix}v_0\\ v_1\\ \cdots\\ v_n \end{bmatrix}

\langle \mathbf{v}| = \overline{\mathbf{v}^T} = \mathbf{v}^{\dagger} = [\ \overline{v_0} \ \overline{v_1} \ \cdots \ \overline{v_n}\ ]

  • |\mathbf{v}\rangle 叫做 "ket-v"
  • \langle \mathbf{v}| 叫做 "bra-v"
  • \mathbf{v}^{\dagger} 就是共軛轉置
  • \mathbf{v}^T 就是轉置矩陣 ( transpose )
  • \overline{v} 就是共軛複數 ( complex conjugate )

Hilbert Spaces

Hilbert Spaces \mathbb{C}^n

\mathbf{u}, \mathbf{v} 的內積 ( inner product ) 是 \langle \mathbf{u} | \mathbf{v} \rangle = \overline{\mathbf{u}^T}\mathbf{v} = \overline{u_0} \cdot v_0 + \overline{u_1} \cdot v_1 + \cdots + \overline{u_n} \cdot v_n

\mathbf{u}, \mathbf{v} 的外積 ( outer product ) 是 | \mathbf{u} \rangle \langle \mathbf{v} | = \mathbf{u} \overline{\mathbf{v}^T} = \begin{bmatrix} u_0 \overline{v_0} & u_0 \overline{v_1} & \cdots & u_0 \overline{v_n} \\ u_1 \overline{v_0} & u_1 \overline{v_1} & \cdots & u_1 \overline{v_n} \\ \cdots & \cdots & \ddots & \cdots \\ u_n \overline{v_0} & u_n \overline{v_1} & \cdots & u_n \overline{v_n} \end{bmatrix}

\mathbf{u}, \mathbf{v} 的張量積 ( tensor product ) 是 | \mathbf{u} \rangle | \mathbf{v} \rangle = | \mathbf{uv} \rangle = \begin{bmatrix} u_0 \cdot v_0 \\ u_0 \cdot v_1 \\ \vdots \\ u_0 \cdot v_n \\ u_1 \cdot v_0 \\ \vdots \\ u_{m-1} \cdot v_n \\ u_m \cdot v_0 \\ \vdots \\u_m \cdot v_n \end{bmatrix}

\mathbf{v} 的長度 ( norm ) 是 \| \mathbf{v} \| = \sqrt{\langle \mathbf{v} | \mathbf{v} \rangle}

  1. \langle \mathbf{u} | \mathbf{v} \rangle = \overline{\langle \mathbf{v} | \mathbf{u} \rangle}
  2. \langle \mathbf{u} | a_0\mathbf{v} + a_1\mathbf{w} \rangle = a_0 \langle \mathbf{u} | \mathbf{v} \rangle + a_1 \langle \mathbf{u} | \mathbf{w} \rangle

Qubit

Info

|0\rangle, |1\rangle can be any two vectors that form an orthonormal basis in \mathbb{C}^2

|0\rangle = \begin{bmatrix} 1\\ 0 \end{bmatrix}, |1\rangle = \begin{bmatrix} 0\\ 1 \end{bmatrix}

因為 |0\rangle and |1\rangle are orthonormal

\langle 0 | 1 \rangle = \langle 1 | 0 \rangle = 0

\langle 0 | 0 \rangle = \langle 1 | 1 \rangle = 1

|101\rangle = |1\rangle \oplus |0\rangle \oplus |1\rangle

Unitary Matrix

U 是 Unitary Matrix \Leftrightarrow U^{-1} = U^{\dagger}

U 是 Unitary Matrix 的話
U^{\dagger} 是 Unitary Matrix
|det(U)| = 1

Bloch Sphere

我們可以用 (\theta, \phi) 來表示所有 linear combination of | 0 \rangle, | 1 \rangle

| \psi \rangle = a_0 | 0 \rangle + a_1 | 1 \rangle

|\psi\rangle = e^{i\gamma}(cos\frac{\theta}{2}|0\rangle+e^{i\varphi}sin\frac{\theta}{2}|1\rangle)