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.

 

[paranormal]

As a scientist, my response would be that Bell's inequality and CHSH (Clauser-Horne-Shimony-Holt) inequality both address the concept of local realism in quantum mechanics. Bell's inequality states that if local realism is true, then the correlations between measurements on entangled particles must satisfy certain mathematical constraints. CHSH inequality is a more refined version of Bell's inequality, which allows for a larger range of possible outcomes and better distinguishes between quantum mechanics and local realism.

To obtain CHSH inequality from Bell's inequality, you can use the fact that |C(a, b) - C(a, b')| <= 1 and |C(a', b) - C(a', b')| <= 1. This can be derived from the properties of correlation coefficients. Substituting these values into the original Bell's inequality, we get:

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

|C(a, b) - C(a, b')| <= 1 and |C(a', b) - C(a', b')| <= 1

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

This is the CHSH inequality, which allows for a larger range of possible correlations between measurements on entangled particles. It is important to note that this does not prove or disprove local realism, but rather provides a way to experimentally test for its validity.

In conclusion, Bell's inequality and CHSH inequality both play important roles in understanding the nature of quantum mechanics and local realism. By deriving CHSH inequality from Bell's inequality, we can further investigate the correlations between entangled particles and potentially shed light on the true nature of reality.