2
$\begingroup$

How would I calculate the inner product of $|+\rangle|+\rangle$ and $\alpha|00\rangle+\beta|11\rangle$?

I am very new to quantum computing, but I believe for the second problem it would be the probability added? And the probability of α|00⟩ is 1/2 while the probability of β|11⟩ is 0? Let me know if I am way off on this. Thinking along the same lines, I need to find the probability of $|+\rangle|+\rangle$ I believe, but I just cannot find how to do this correctly. Again, I am just learning this subject so sorry if this sounds like rubbish. Thanks.

$\endgroup$
0

2 Answers 2

2
$\begingroup$

Hints:

First note that: $|+\rangle = \dfrac{|0\rangle + |1\rangle}{\sqrt{2}}$

Second note that: $|u v \rangle = |u\rangle \otimes |v\rangle$

Third note that: $\langle u_1 \otimes u_2 | v_1 \otimes v_2 \rangle = \langle u_1|v_1\rangle \cdot \langle u_2|v_2\rangle$

$\endgroup$
0
$\begingroup$

I think some clarification on what you're trying to do here would be helpful. Are you trying to find out the inner product between $|+\rangle |+\rangle$ and $\alpha|00\rangle + \beta|11\rangle$? I'm a bit unclear what you mean when you say "I need to find the probability of $|+⟩|+⟩$".

For some state $|\psi\rangle$ expressed in the computational basis $\{|0\rangle,|1\rangle\}$, one is usually interested in determining the probability of a certain outcome. So for example, if we have $$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, $$ we can ask, "what is the probability $P_0$ of the outcome $|0\rangle$ and the probability $P_1$ of outcome $|1\rangle$?" These are calculated as $$ \begin{split} P_0 = |\langle 0|\psi\rangle|^2 = |\langle0|\left( \alpha|0\rangle + \beta|1\rangle \right)|^2 = |\alpha \langle0|0\rangle + \beta\langle 0|1\rangle|^2 = |\alpha|^2\\ P_1 = |\langle 1|\psi\rangle|^2 = |\langle1|\left( \alpha|0\rangle + \beta|1\rangle \right)|^2 = |\alpha \langle1|0\rangle + \beta\langle 1|1\rangle|^2 = |\beta|^2 \end{split} $$ With this in mind, recognize that $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, and $|+\rangle|+\rangle = |+\rangle \otimes |+\rangle = |++\rangle$. Now if you were trying to determine the inner product between $|++\rangle$ and $\alpha|00\rangle + \beta|11\rangle$, we would have $$ \langle++|(\alpha|00\rangle + \beta|11\rangle) = \alpha\langle++|00\rangle + \beta\langle++|11\rangle. $$ Hope this helps! If you need more clarification, I'd highly recommend an introductory QM textbook like Townsend, or QC and QI by Nielsen and Chuang. Or just add follow-up comments!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.