I try to follow the calculation from IBMQ experience regrading of Entanglement and Bell test which they derive the value of question as \begin{equation} C=\langle A B\rangle-\left\langle A B^{\prime}\right\rangle+\left\langle A^{\prime} B\right\rangle+\left\langle A^{\prime} B^{\prime}\right\rangle \end{equation} where ; $\langle A B\rangle= p_{++}+p_{--}-p_{+-}-p_{-+}$ and $A , A'$ is Alice's basis, $B, B'$ is Bob's basis of measurement

the choice of the basis they chose is following this $A = Z, A' = X$ and $B = W, B' = V$ where The Bell test with quantum correlations The required measurements for the CHSH experiment

then they have \begin{equation} \langle Z W\rangle=\langle Z V\rangle=\langle X W\rangle=\frac{1}{\sqrt{2}},\langle X V\rangle=-\frac{1}{\sqrt{2}} \end{equation} so that $|C| = 2\sqrt{2}$ which greater than $2$

Now, they do this experiment on the real device and obtain the following value

Example result

The question is if following their choice of basis then $\langle AB' \rangle$ = $\langle ZV' \rangle$ but from their calculation, it can be seen that $\langle AB' \rangle$ = $\langle XV' \rangle$ which seem to agree with above value.

So what is supposed to be right and wrong in this calculation and how can I know theoretically which basis is supposed to have a minus sign from this choice of basis.

Note : the link to reference will redirect to another page https://quantum-computing.ibm.com/support/guides/introduction-to-quantum-circuits?page=5cae705d35dafb4c01214bc5 from that page --> Introduction to Quantum Circuits --> Entanglement and Bell Tests Sorry for the inconvenience.


1 Answer 1


The fact is that it doesn't matter. So long as one of the four terms has the minus sign on it, you'll get a suitable test. Yes, there seems to be some inconsistency with labelling the different cases, but it's not a big deal - nothing physically changes if you swap which measurement basis you call $A$ and which you call $A'$ for example; you're still doing the same experiment. Obviously in this case you want to calculate $$ \langle ZW\rangle+\langle ZV\rangle+\langle XW\rangle-\langle XV\rangle. $$


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