Quantum mechanics was discovered around 1920, but Bohr and Einstein thoroughly and intensively debated over the merits of a probabilistic interpretation of quantum mechanics. In the end, Einstein’s theory was rejected, and the theory of wave functions (the wave-particle duality) based on Bohr’s probabilistic interpretation has explained many experimental facts. Therefore, the theory based on local interactions supported by Einstein is incorrect, and the theory based on long-distance interactions, in which correlations between particles far apart from each other are transmitted by the contraction of the wave function, has explained many experimental facts.
Now, let us introduce a thought experiment on the correlation of two particles based on a long-distance interaction (quantum entanglement). To do so, let us consider the case where the state of two particles (the two-particle state) can take four orthogonal states called the Bell states .
\(\displaystyle \Ket{\varPsi^{(\pm)}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\downarrow_{2}} \pm \Ket{\downarrow_{1}}\otimes\Ket{\uparrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} \pm \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right) \),
\(\displaystyle \Ket{\varPhi^{(\pm)}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\uparrow_{2}} \pm \Ket{\downarrow_{1}}\otimes\Ket{\downarrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} \pm \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\).
Next, we give the two-particle EPR state as follows:
\(\displaystyle \Ket{\varphi^{\mathrm{EPR}}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\otimes\Ket{\uparrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right) =\Ket{\varPsi^{(-)}} \).
Finally, consider an isolated single-particle state described by a superposition of two different states (\(\Ket{\uparrow}\) and \(\Ket{\downarrow}\)):
\(\displaystyle \Ket{\varphi^{\mathrm{Isolated}}} = C_{1}\Ket{\uparrow} + C_{2}\Ket{\downarrow} \).
here, \(C_{1}\) and \(C_{1}\) are the coefficients of the single-particle-state superposition.
Next, we will show that the state of three particles (the three-particle state) can be entangled by the direct product of the one-particle state and the two-particle EPR state . The three-particle state can be written as follows:
\(\displaystyle
\begin{split}
\Ket{\varphi^{\mathrm{Entangled}}}
=&\Ket{\varphi^{\mathrm{Isolated}}}\otimes\Ket{\varphi^{\mathrm{EPR}}} \\
=&\Ket{\varphi^{\mathrm{Isolated}}}\Ket{\varphi^{\mathrm{EPR}}} \\
=&\left(C_{1}\Ket{\uparrow} + C_{2}\Ket{\downarrow}\right)\otimes\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{2}}\Ket{\downarrow_{3}} – \Ket{\downarrow_{2}}\Ket{\uparrow_{3}}\right)\right]\\
=&
\dfrac{1}{2}
\left\{
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} – \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\right]
\otimes
\left(C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right) \right. \\
&+
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} + \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\right]
\otimes
\left(C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right)\right]
\otimes
\left(-C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\left.
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} + \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right)\right]
\otimes
\left(-C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)
\right\} \\
=&
\Ket{\varPhi^{(-)}}\otimes\left(C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)\\
&+
\Ket{\varPhi^{(+)}}\otimes\left(C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right)\\
&+
\Ket{\varPsi^{(-)}}\otimes\left(-C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\Ket{\varPsi^{(+)}}\otimes\left(-C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)
\end{split}
\)
Therefore, when one Bell state (the two-particle state) is observed, the other single-particle state is uniquely determined. Since the other single-particle state is uniquely determined, it has been demonstrated that quantum teleportation can be realistic.
1920年頃に量子力学が構築されたが，量子力学の確率的解釈の是非をめぐってボーアとアインシュタインは論争を繰り広げた．結局，アインシュタインの指示した理論は棄却され，ボーアの確率的解釈に基づく波動関数の理論（波と粒子のニ重性）は多くの実験事実を説明してきた．従って，アインシュタインが支持した局所的相互作用に基づく理論は誤っており，遠隔的相互作用に基づいてお互いが遠く離れた粒子間の相関が波動関数の収縮により伝わる理論が多くの実験事実を説明してきた．
さて，遠隔的相互作用に基づく2個の粒子の相関（量子もつれ）に関する思考実験を紹介しよう．そのために，2個の粒子状態（2粒子状態）がベル状態と呼ばれる直交する4種類の状態を取る場合を考えよう．
\(\displaystyle \Ket{\varPsi^{(\pm)}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\downarrow_{2}} \pm \Ket{\downarrow_{1}}\otimes\Ket{\uparrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} \pm \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right) \),
\(\displaystyle \Ket{\varPhi^{(\pm)}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\uparrow_{2}} \pm \Ket{\downarrow_{1}}\otimes\Ket{\downarrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} \pm \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\).
次に，下記のように2粒子EPR状態を与えよう：
\(\displaystyle \Ket{\varphi^{\mathrm{EPR}}} = \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\otimes\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\otimes\Ket{\uparrow_{2}}\right)
= \dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right) =\Ket{\varPsi^{(-)}} \).
最後に，2つの異なる状態（\(\Ket{\uparrow}\)と\(\Ket{\downarrow}\)）の重ね合わせで記述される孤立した1粒子状態を考えよう：
\(\displaystyle \Ket{\varphi^{\mathrm{Isolated}}} = C_{1}\Ket{\uparrow} + C_{2}\Ket{\downarrow} \).
ここで，\(C_{1}\)と\(C_{2}\)は重ね合わせの係数である．
次に1粒子状態と2粒子EPR状態の直積で，3個の粒子状態（3粒子状態）が量子もつれを引き起こすことを示そう．その3粒子状態は以下のように記述される：
\(\displaystyle
\begin{split}
\Ket{\varphi^{\mathrm{Entangled}}}
=&\Ket{\varphi^{\mathrm{Isolated}}}\otimes\Ket{\varphi^{\mathrm{EPR}}} \\
=&\Ket{\varphi^{\mathrm{Isolated}}}\Ket{\varphi^{\mathrm{EPR}}} \\
=&\left(C_{1}\Ket{\uparrow} + C_{2}\Ket{\downarrow}\right)\otimes\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{2}}\Ket{\downarrow_{3}} – \Ket{\downarrow_{2}}\Ket{\uparrow_{3}}\right)\right]\\
=&
\dfrac{1}{2}
\left\{
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} – \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\right]
\otimes
\left(C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right) \right. \\
&+
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\uparrow_{2}} + \Ket{\downarrow_{1}}\Ket{\downarrow_{2}}\right)\right]
\otimes
\left(C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} – \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right)\right]
\otimes
\left(-C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\left.
\left[\dfrac{1}{\sqrt{2}}\left(\Ket{\uparrow_{1}}\Ket{\downarrow_{2}} + \Ket{\downarrow_{1}}\Ket{\uparrow_{2}}\right)\right]
\otimes
\left(-C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)
\right\} \\
=&
\Ket{\varPhi^{(-)}}\otimes\left(C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)\\
&+
\Ket{\varPhi^{(+)}}\otimes\left(C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right)\\
&+
\Ket{\varPsi^{(-)}}\otimes\left(-C_{1}\Ket{\uparrow_{3}} – C_{2}\Ket{\downarrow_{3}}\right) \\
&+
\Ket{\varPsi^{(+)}}\otimes\left(-C_{1}\Ket{\uparrow_{3}} + C_{2}\Ket{\downarrow_{3}}\right)
\end{split}
\)
従って，1つのベル状態（2粒子状態）が観測されたら，他方の1粒子状態は一意に定められる．他方の1粒子状態が一意に定められたので，量子テレポーテーションが実現可能であることが示された．
