# In Bell are pairs independent?

1. Aug 16, 2015

### jk22

let A B B' be Bell usual matrices for spin 1/2.

Are the results of $$A\otimes B$$ and $$A\otimes B'$$ independent ?

I cannot imagine the contrary since on what could they depend all angles are fixed.

2. Aug 16, 2015

### Strilanc

You need to be clearer; give more context.

What exactly are the matrices $B$ and $B'$? Googling "Bell matrix" brings up nothing obviously applicable. And what is $A$? And what results are you talking about? There's fixed angles somewhere?

You're probably asking something about tests of Bell inequalities, but I'm not sure what exactly.

3. Aug 21, 2015

### jk22

A and B are the projection of the spin operator along direction of measurement : $$A=\vec{\sigma}\cdot\vec{n}_A$$

All matrices are defined the same way, we can take an angle in the x z plane for exsmple.

Now the result of measurement of $$C=A\otimes B$$ is either 1 or -1. My question is, if we define $$C'=A\otimes B'$$ : do we have p(C=1,C'=1)=p(C=1)p(C'=1) ? Or are those dependent ?

4. Aug 21, 2015

### Strilanc

I don't think the probability p(C=1, C'=1) is well defined in this case, because the measurements may not commute.

For example, for the bell pair $\left| 00 \right\rangle + \left| 11 \right\rangle$, the result of $X \otimes X$ is always the same so you don't affect the system whereas the result of $X \otimes Z$ is 50/50ish and does mess up the system. So if you measure $X \otimes X$ then $X \otimes Z$ you'll get both =1 about 50% of the time, but if you measure $X \otimes Z$ then $X \otimes X$ it will drop to 25%.

So at the very least you must be explicit about the measurement order, I think.

5. Aug 21, 2015

### jk22

In the notation p(a=1,b=1) the order is important since the function is not forcedly symmetrical. However, The measurement are supposed to be simultaneous or better said with the same initial state so that in that case i suppose the order is not relevant

Last edited: Aug 21, 2015
6. Aug 21, 2015

### Avodyne

You cannot measure B and B' simultaneously. So, in QM, p(B,B') is not defined.

In hidden variable theory, p(B,B') is defined. That's essentially what it means to have a hidden variable theory.

7. Aug 22, 2015

### jk22

So in qm we could define measuring B before B' ?

What about if we pass the photon to a half-silvered mirror and send half of it to B and the other to B' ?

8. Aug 22, 2015

### Staff: Mentor

You could, but then P(B') is fully determined by the measurement result of B and has nothing to do with the entanglement any more.
Then you'll detect a photon "at B" or "at B'" - half of the time you perform one experiment, half of the time another, but you never get two measurements of the same photon then.

9. Aug 22, 2015

### jk22

So passing through a beam splitter does not make the photon go in superposition of both path ?

10. Aug 22, 2015

### Staff: Mentor

It does, but measuring it later means you only observe one path.

11. Aug 26, 2015

### gill1109

Your observables C and C' do not commute so they cannot be measured at the same time.