How Can Bell's Inequality Be Transformed into the CHSH Inequality?

[cscott]

Homework Statement

I need to get the CHSH inequality from Bell's inequality

Homework Equations

|C(a, b) - C(a, b')| + |C(a', b) - C(a', b')| <= 2

to

-2 <= C(a, b) - C(a, b') + C(a', b) + C(a', b') <= 2

The Attempt at a Solution

I know the CHSH allows for no correlation of 0 but I can't get anywhere. I think my math with inequalities just sucks.

 

[Gordon Watson]

Homework Statement

I need to get the CHSH inequality from Bell's inequality

Homework Equations

|C(a, b) - C(a, b')| + |C(a', b) - C(a', b')| <= 2

to

-2 <= C(a, b) - C(a, b') + C(a', b) + C(a', b') <= 2

The Attempt at a Solution

I know the CHSH allows for no correlation of 0 but I can't get anywhere. I think my math with inequalities just sucks.

The above looks a bit confused. How about this trick?

(1) (a + a')b + (a - a')b' = +/-2;

since (a + a') = 0 AND (a - a') = +/-2, XOR (a + a') = +/-2 AND (a - a') = 0.

So, expanding (1):

(2) ab + a'b + ab' - a'b' = +/-2.

Then, ensemble-averaging:

(3) <ab> + <a'b> + <ab'> - <a'b'> </= +/-2.

Or:

(4) |<ab> + <a'b> + <ab'> - <a'b'>| </= 2;

where <ab> = C(ab), etc. (4) is known as the CHSH Inequality; and the real trick is to see why EPR-Bell tests cannot (in general) satisfy (4).

 

[mmmrrrrrrr]

The above looks a bit confused. How about this trick?

(1) (a + a')b + (a - a')b' = +/-2;

since (a + a') = 0 AND (a - a') = +/-2, XOR (a + a') = +/-2 AND (a - a') = 0.

So, expanding (1):

(2) ab + a'b + ab' - a'b' = +/-2.

Then, ensemble-averaging:

(3) <ab> + <a'b> + <ab'> - <a'b'> </= +/-2.

Or:

(4) |<ab> + <a'b> + <ab'> - <a'b'>| </= 2;

where <ab> = C(ab), etc. (4) is known as the CHSH Inequality; and the real trick is to see why EPR-Bell tests cannot (in general) satisfy (4).

Very nice answer, sir. Now I'd like to ask what you lead to: why cannot the EPR state satisfy the Bell inequality generally?

 

[Gordon Watson]

Very nice answer, sir. Now I'd like to ask what you lead to: why cannot the EPR state satisfy the Bell inequality generally?

Good question, but it's a homework question at our school. So please follow the PF homework rules.