I'm looking at some examples, but I cannot get the expected result when it comes down to making the measurement on the following state where we measure the first qubit which is the ancilla state.

Here is $|\psi\rangle = \frac{1}{2} |{0}\rangle \otimes (|\psi_a\rangle |\psi_b\rangle + |\psi_b\rangle |\psi_a\rangle) + \frac{1}{2} |{1}\rangle \otimes (|\psi_a\rangle |\psi_b\rangle - |\psi_b\rangle |\psi_a\rangle)$

My calculation suggests that the probability of the ancilla being in the state $|0\rangle$ is:

$p_{0} =\frac{1}{2} + \frac{1}{2}|\langle\psi_a | \psi_b \rangle|^2 $

However the text suggests that there is a minus in place of the plus. I'm not sure if I'm doing anything wrong here or this is a typo.

  • $\begingroup$ Does the text say that $p_{0} =\frac{1}{2} - \frac{1}{2}|\langle\psi_a | \psi_b \rangle|^2 $ or $p_{1} =\frac{1}{2} - \frac{1}{2}|\langle\psi_a | \psi_b \rangle|^2 $ $\endgroup$ – Victory Omole Feb 19 '20 at 21:27
  • $\begingroup$ $p_{0}$ it says. I can see a minus coming in if its $p_{1}$. $\endgroup$ – disruptive Feb 20 '20 at 7:21

Your calculation is correct, given the state and conclusion that you have written. The book may be unclear, or have a typo.


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