前提知識

$$\gdef\vec#1{\bm{#1}} \gdef\bra#1{\left\lang#1\right|} \gdef\ket#1{\left|#1\right\rang} \gdef\braket#1#2{\left\lang#1\middle|#2\right\rang} \gdef\abs#1{\left|#1\right|} \gdef\BS#1{\mathrm{B}_{#1}} \gdef\R{\Reals} \gdef\C{\Complex} \gdef\U{U} \gdef\V{V} \gdef\W{W} \gdef\X{X} \gdef\Y{Y} \gdef\e{\vec{0}} \gdef\ii{\mathrm{i}} \gdef\Hilb{\mathbf{Hilb}} \gdef\Hom#1#2{\mathrm{Hom}(#1,#2)} \gdef\op{\mathrm{op}} \gdef\Obj#1{\mathrm{Obj}(#1)} \gdef\Mor#1{\mathrm{Mor}(#1)} \gdef\id#1{\mathrm{id}_{#1}} \gdef\basis#1#2{\vec{#1}_{#2}} \gdef\coeff#1#2{#1^{#2}} \gdef\norm#1{\left\|#1\right\|} \gdef\dist{d} \gdef\pauli#1#2{\sigma_{#1}^{#2}} \gdef\bbra#1#2{\vphantom{\bra{#2}}_{#1\!}\bra{#2}} \gdef\bket#1#2{\ket{#2}_{#1}} \gdef\su{\uparrow} \gdef\sd{\downarrow}$$

ディラック記法

２次元複素ヒルベルト空間$\V_{i}$を考える。
複素数体を$\C$、正規直交基底を$\{\vec{\su}, \vec{\sd}\} \subset \V_{i}$とする。

ブラ

内積$\cdot \colon \V_{i} \times \V_{i} \longrightarrow \C$により
線型写像$\bbra{i}{\phi} \equiv (\vec{\phi} \cdot -) \colon \V_{i} \longrightarrow \C$を定義すれば
$\bbra{i}{\phi} \colon \vec{x} \longmapsto \vec{\phi} \cdot \vec{x}$となる。

ケット

スカラー乗法$\cdot \colon \V_{i} \times \C \longrightarrow \V_{i}$により
線型写像$\bket{i}{\psi} \equiv (\vec{\psi} \cdot -) \colon \C \longrightarrow \V_{i}$を定義すれば
$\bket{i}{\psi} \colon c \longmapsto \vec{\psi} \cdot c$となる。

ブラ・ケット

合成写像$\bbra{i}{\phi} \circ \bket{i}{\psi} \colon \C \longrightarrow \C$は
$\bbra{i}{\phi} \circ \bket{i}{\psi} \colon c \longmapsto \vec{\phi} \cdot (\vec{\psi} \cdot c) = (\vec{\phi} \cdot \vec{\psi}) \cdot c$となる。

ケット・ブラ

合成写像$\bket{i}{\psi} \circ \bbra{i}{\phi} \colon \V_{i} \longrightarrow \V_{i}$は
$\bket{i}{\psi} \circ \bbra{i}{\phi} \colon \vec{x} \longmapsto \vec{\psi} \cdot (\vec{\phi} \cdot \vec{x})$となる。

パウリ行列

線型写像$\pauli{i}{\lambda} \colon \V_{i} \longrightarrow \V_{i}$を
ケット・ブラを用いて以下のように定義する。

$$\begin{aligned} \pauli{i}{e} &\equiv \bket{i}{\su}\circ\bbra{i}{\su} + \bket{i}{\sd}\circ\bbra{i}{\sd} = \id{\V_{i}} \\ \pauli{i}{z} &\equiv \bket{i}{\su}\circ\bbra{i}{\su} - \bket{i}{\sd}\circ\bbra{i}{\sd} \\ \pauli{i}{x} &\equiv \bket{i}{\su}\circ\bbra{i}{\sd} + \bket{i}{\sd}\circ\bbra{i}{\su} \\ \pauli{i}{y}\cdot\ii &\equiv \bket{i}{\su}\circ\bbra{i}{\sd} - \bket{i}{\sd}\circ\bbra{i}{\su} \end{aligned}$$

恒等写像$\id{\V_{i}} \colon \V_{i} \longrightarrow \V_{i}$は
$\id{\V_{i}} \colon \vec{x} \longmapsto \vec{x}$となる。

ベル状態

線型写像$\bket{i,j}{\phi^{\pm}} \colon \C \longrightarrow \V_{i} \otimes \V_{j}$および
線型写像$\bket{i,j}{\psi^{\pm}} \colon \C \longrightarrow \V_{i} \otimes \V_{j}$は
線型写像のテンソル積を用いて以下のように記述できる。

$$\begin{aligned} \bket{i,j}{\phi^{+}} &\cong \left(\bket{i}{\su}\otimes\bket{j}{\su} + \bket{i}{\sd}\otimes\bket{j}{\sd}\right) \cdot \frac{1}{\sqrt{2}} \cong (\pauli{i}{e} \otimes \id{\V_{j}}) \circ \bket{i,j}{\phi^{+}} \\ \bket{i,j}{\phi^{-}} &\cong \left(\bket{i}{\su}\otimes\bket{j}{\su} - \bket{i}{\sd}\otimes\bket{j}{\sd}\right) \cdot \frac{1}{\sqrt{2}} \cong (\pauli{i}{z} \otimes \id{\V_{j}}) \circ \bket{i,j}{\phi^{+}} \\ \bket{i,j}{\psi^{+}} &\cong \left(\bket{i}{\su}\otimes\bket{j}{\sd} + \bket{i}{\sd}\otimes\bket{j}{\su}\right) \cdot \frac{1}{\sqrt{2}} \cong (\pauli{i}{x} \otimes \id{\V_{j}}) \circ \bket{i,j}{\phi^{+}} \\ \bket{i,j}{\psi^{-}} &\cong \left(\bket{i}{\su}\otimes\bket{j}{\sd} - \bket{i}{\sd}\otimes\bket{j}{\su}\right) \cdot \frac{1}{\sqrt{2}} \cong (\pauli{i}{y}\cdot\ii \otimes \id{\V_{j}}) \circ \bket{i,j}{\phi^{+}} \end{aligned}$$

量子テレポーテーション

線型写像$\left(\id{\V_{0}} \otimes \bket{1,2}{\phi^{+}}\right) \colon \V_{0} \otimes \C \longrightarrow \V_{0} \otimes \V_{1} \otimes \V_{2}$と
線型写像$\left(\bbra{0,1}{\phi^{+}} \otimes \id{\V_{2}}\right) \colon \V_{0} \otimes \V_{1} \otimes \V_{2} \longrightarrow \C \otimes \V_{2}$の
合成写像$\left(\bbra{0,1}{\phi^{+}} \otimes \id{\V_{2}}\right) \circ \left(\id{\V_{0}} \otimes \bket{1,2}{\phi^{+}}\right) \colon \V_{0} \otimes \C \longrightarrow \C \otimes \V_{2}$は

$$\begin{aligned} &\hspace{-3mm} \left(\bbra{0,1}{\phi^{+}} \otimes \id{\V_{2}}\right) \circ \left(\id{\V_{0}} \otimes \bket{1,2}{\phi^{+}}\right) \cdot 2 \\ &\cong \left(\bbra{0}{\su}\otimes\bbra{1}{\su} \otimes \id{\V_{2}} + \bbra{0}{\sd}\otimes\bbra{1}{\sd} \otimes \id{\V_{2}}\right) \circ \left(\id{\V_{0}} \otimes \bket{1}{\su}\otimes\bket{2}{\su} + \id{\V_{0}} \otimes \bket{1}{\sd}\otimes\bket{2}{\sd}\right) \\ &= \bbra{0}{\su}\otimes\id{\C}\otimes\bket{2}{\su} + \bbra{0}{\sd}\otimes\id{\C}\otimes\bket{2}{\sd} \\ &\cong \bket{2}{\su}\otimes\bbra{0}{\su} + \bket{2}{\sd}\otimes\bbra{0}{\sd} \end{aligned}$$

となり、これは、状態$\bket{0}{v} \equiv \bket{0}{\su}\cdot\alpha + \bket{0}{\sd}\cdot\beta$を状態$\bket{2}{v} \equiv \bket{2}{\su}\cdot\alpha + \bket{2}{\sd}\cdot\beta$に移す。

$$\begin{aligned} &\hspace{-3mm} \overbrace{ \vphantom{\left(\bbra{0,1}{\phi^{+}}\bket{1,2}{\phi^{+}}\right)} \left(\id{\C} \otimes \pauli{2}{\lambda}\right) }^{\text{復元}} \circ \overbrace{ \left( \bigl(\bbra{0,1}{\phi^{+}} \circ (\pauli{0}{\lambda}\otimes\id{\V_{1}})\bigr)\otimes\id{\V_{2}} \right) \cdot 2 }^{\text{ベル測定}} \circ \overbrace{ \left(\bket{0}{v} \otimes \bket{1,2}{\phi^{+}}\right) }^{\text{初期状態}} \\ &= \left(\id{\C} \otimes \pauli{2}{\lambda}\right) \circ \left( \bbra{0,1}{\phi^{+}} \otimes\id{\V_{2}}\right) \circ \left(\id{\V_{0}} \otimes \bket{1,2}{\phi^{+}}\right) \cdot 2 \circ \left(\left(\pauli{0}{\lambda} \circ \bket{0}{v}\right) \otimes \id{\C}\right) \\ &\cong \left(\pauli{2}{\lambda}\otimes \id{\C}\right) \circ \left(\bket{2}{\su}\otimes\bbra{0}{\su} + \bket{2}{\sd}\otimes\bbra{0}{\sd}\right) \circ \left(\id{\C} \otimes \left(\pauli{0}{\lambda} \circ \bket{0}{v}\right)\right) \\ &= \underbrace{\left(\left(\pauli{2}{\lambda}\circ\pauli{2}{\lambda}\right) \otimes \id{\C} \right)}_{\id{\V_{2}} \otimes \ \id{\C}} \circ \left(\bket{2}{\su}\otimes\bbra{0}{\su} + \bket{2}{\sd}\otimes\bbra{0}{\sd}\right) \circ \left(\id{\C} \otimes \bket{0}{v}\right) \\ &= \left(\bket{2}{\su}\otimes\bbra{0}{\su} + \bket{2}{\sd}\otimes\bbra{0}{\sd}\right) \circ \left(\id{\C} \otimes \bket{0}{v}\right) \\ &= \left(\bket{2}{\su}\cdot\alpha + \bket{2}{\sd}\cdot\beta\right)\otimes\id{\C} \cong \bket{2}{v} \end{aligned}$$

参考文献

ウィキペディア「量子テレポーテーション」(2020-07-14)