Quantum Information Science I メモその10 内積

内積(Inner product)

普通の内積 $\def\bra#1{\mathinner{\left\langle{#1}\right|}} \def\ket#1{\mathinner{\left|{#1}\right\rangle}} \def\braket#1#2{\mathinner{\left\langle{#1}\middle|#2\right\rangle}}\\ \require{HTML}$ $\displaystyle \braket{\psi}{\phi}=\sum_{00}^{11}\psi^{*}_i{\phi}_i$

$\displaystyle \psi_1 = \ket{\phi_1}\otimes\ket{\phi_2}$ , $\displaystyle \psi_2 = \ket{\phi_3}\otimes\ket{\phi_4}$

とすると

$\displaystyle \braket{\psi_1}{\psi_2}=\braket{\phi_1}{\phi_3}\braket{\phi_2}{\phi_4}$

証明というか、計算は

$\displaystyle \phi_1=(\alpha_1\ket{0}+\beta_1\ket{1})$

$\displaystyle \phi_2=(\gamma_1\ket{0}+\delta_1\ket{1})$

$\displaystyle \phi_3=(\alpha_2\ket{0}+\beta_2\ket{1})$

$\displaystyle \phi_4=(\gamma_2\ket{0}+\delta_2\ket{1})$

とすると

$\displaystyle \psi_1= \ket{\phi_1}\otimes\ket{\phi_2} = (\alpha_1\ket{0}+\beta_1\ket{1})(\gamma_1\ket{0}+\delta_1\ket{1})$

$\displaystyle \psi_2= \ket{\phi_3}\otimes\ket{\phi_4} = (\alpha_2\ket{0}+\beta_2\ket{1})(\gamma_2\ket{0}+\delta_2\ket{1})$

$\displaystyle \braket{\psi_1}{\psi_2}=\alpha_1^{\ast} \gamma_1^{\ast}\alpha_2 \gamma_2 + \alpha_1^{\ast} \delta_1^{\ast} \alpha_2 \delta_2 + \beta_1^{\ast} \gamma_1^{\ast}\beta_2 \gamma_2 + \beta_1^{\ast} \delta_1^{\ast} \beta_2 \delta_2$