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『Quantum Computing: A Gentle Introduction』の演習問題を解く 3.7

Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation) (English Edition)

Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation) (English Edition)

目次はこちら

Exercise 3.7. Write the following states in terms of the Bell basis.

  • a.  { |00\rangle }
  • b.  { |+\rangle|-\rangle }
  • c.  { \frac{1}{\sqrt{3}}\left(|00\rangle + |01\rangle + |10\rangle\right) }

補足
ベル状態とは以下の4つの状態:

  { \displaystyle\begin{align*}
  |\Phi^+\rangle &= \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right) \\
  |\Phi^-\rangle &= \frac{1}{\sqrt{2}}\left(|00\rangle - |11\rangle\right) \\
  |\Psi^+\rangle &= \frac{1}{\sqrt{2}}\left(|01\rangle + |10\rangle\right) \\
  |\Psi^-\rangle &= \frac{1}{\sqrt{2}}\left(|01\rangle - |10\rangle\right)
\end{align*}}

ベル基底とはこれら4つの状態からなる 2-Qubit 系の基底です。

a.
  { \displaystyle\begin{align*}
  |00\rangle &= \frac{1}{\sqrt{2}}\left(|\Psi^+\rangle + |\Psi^-\rangle\right)
\end{align*}}

b.
  { \displaystyle\begin{align*}
  |+\rangle|-\rangle
    &= \frac{1}{2}\left(|0\rangle + |1\rangle\right)\left(|0\rangle - |1\rangle\right) \\
    &= \frac{1}{2}\left(|00\rangle - |01\rangle + |10\rangle - |11\rangle\right) \\
    &= \frac{1}{\sqrt{2}}\left(|\Phi^-\rangle - |\Psi^-\rangle\right)
\end{align*}}

c.
  { \displaystyle\begin{align*}
  \frac{1}{\sqrt{3}}\left(|00\rangle + |01\rangle + |10\rangle\right)
    &= \frac{1}{\sqrt{3}}\left\{\frac{1}{\sqrt{2}}\left(|\Phi^+\rangle + |\Phi^-\rangle\right) + \sqrt{2}|\Psi^+\rangle\right\} \\
    &= \frac{1}{\sqrt{6}}\left(|\Phi^+\rangle + |\Phi^-\rangle + 2|\Psi^+\rangle\right)
\end{align*}}