I'm trying to prove Bell's inequality by following the one made by Wigner, that is:
$P(+_a,+_b) \le P(+_a,+_c) + P(+_c,+_b)$
And that this proof is NOT satisfied for angles between the various axes $\hat a$, $\hat b$ and $\hat c$. In particular I have chosen the coplanar axes (plane x-y) and that the $\hat c$ axis is the bisector of $\hat a$ and $\hat b$.
So I'm in the situation in the picture Graph
My idea is to take the singlet state (represented in this case by a specific state of the Bell state) and measure one qubit on one axis ($\hat a$, $\hat b$, $\hat c$) and the other qubit on one of the remaining axes in in such a way as to make all possible combinations.
In the specific situation represented in the graph, the angle that I can choose between the various axes must be a value between $0$ and $\frac \pi2$.
Now I come to my question: my doubt is whether it is correct how to interpret the measurement phase after I have created the singlet state (entagled state).
The entangled state I create it like this:
circ = QuantumCircuit(2,2) circ.x(0) circ.x(1) circ.h(0) circ.cx(0,1)
Referring to the graph I put earlier, if I wanted to now measure a long qubit $\hat a$ and a long $\hat b$ I would have to measure along x (no rotation for the first case) and rotate the second theta qubit around z and then measure along x, correct?
I thought about doing it this way
# Long a-axis for qubit 1 circ.h(0) # Long b-axis for qubit 2 circ.rz(pi/3, 1) circ.h(1) circ.measure(range(2),range(2))
I wonder if these steps are correct or I forgot something or I can do it in another way. Because if I measured directly without applying the Hadamard operator before the measurement I would make a measurement along the z-axis I suppose, right?
I hope I was clear enough and that we can resolve my doubts! Thank you for your time!