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I'm slightly confused by the solution provided below by a suggested solution online to convert |$\phi^+$⟩ to |$\psi^-$⟩.

I tried doing the operation XZ but I got $\frac{1}{\sqrt2}$(|10⟩-|01⟩) instead of |$\psi^-$⟩.

However, applying ZX seems to provide me with the right answer.

Would appreciate the verification!

$$ \begin{align} (XZ \otimes I) |\Phi^+\rangle &= \frac{1}{\sqrt{2}}(XZ|0\rangle \otimes I |0\rangle + XZ |1\rangle \otimes |1\rangle) \\ &= \frac{1}{\sqrt{2}}(X|0\rangle \otimes I|0\rangle - X|1\rangle \otimes I |1\rangle) \\ &= \frac{1}{\sqrt{2}}(|1\rangle \otimes |0\rangle - |0\rangle \otimes |1\rangle) \\ &= |\Psi^-\rangle \end{align} $$

epelaez
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Colin Hong
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1 Answers1

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@Kenneth comment is right.

Note that:

$$ XZ |\Phi^+\rangle = XZ\bigg(\dfrac{|00\rangle + |11\rangle}{\sqrt{2}} \bigg) = \dfrac{|10\rangle - |01\rangle}{\sqrt{2}} = - \bigg( \dfrac{|01\rangle - |10\rangle}{\sqrt{2}} \bigg) = -|\Psi^-\rangle $$

But there is no distinction between the state $-|\Psi^-\rangle $ and $|\Psi^-\rangle $ quantum mechanically. That is, they are equivalent.

KAJ226
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  • Thanks! How can we prove that these two states are not distinguishable quantum mechanically? – Colin Hong Jun 14 '21 at 02:19
  • @Colin Hong In term of why overall phase doesn't matter, check out this answer here: https://quantumcomputing.stackexchange.com/a/9630/9858 – KAJ226 Jun 14 '21 at 04:44