CHSH expression

Here’s the step-by-step breakdown of the CHSH expression 

The Big Picture:
Imagine Alice and Bob each have a special box connected by a mysterious link (like entangled particles). Each box has two buttons (settings or measurement choices) and gives a result of either +1 or -1 (like spin up or down).

1. Alice's Buttons: A₀ and A₁

2. Bob's Buttons: B₀ and B₁

3. The E(...) Terms – Correlation Scores:

   • E(A₀, B₀) asks: "When Alice presses A₀ and Bob presses B₀ at the same time, how often do their boxes flash the same number (+1 and +1 or -1 and -1) versus different numbers (+1 and -1 or -1 and +1)?"
   • It calculates an average agreement score:
     • +1 means perfect agreement (always same result)
     • 0 means no correlation (results random and independent)
     • -1 means perfect disagreement (always opposite results)
   • So E(A₀, B₀) is the "agreement score" when Alice uses A₀ and Bob uses B₀
   • Similarly:
     • E(A₀, B₁) is the agreement score for A₀ and B₁
     • E(A₁, B₀) is the agreement score for A₁ and B₀
     • E(A₁, B₁) is the agreement score for A₁ and B₁

4. The S Expression – The Combined Score:

   • S = E(A₀, B₀) + E(A₀, B₁) + E(A₁, B₀) - E(A₁, B₁)
   • Translation: Add up the agreement scores for the first three combinations (A₀B₀, A₀B₁, A₁B₀), but subtract the agreement score for the last combination (A₁B₁)

Why This Specific Combination – The Key Insight:

• Classical World (Local Realism):
  If the boxes pre-agreed on answers (like hidden instructions) or only communicate at light-speed, there’s a fundamental limit to how high S can be. No matter how cleverly the boxes are programmed beforehand, S can never be larger than 2. So S must be less than or equal to 2.

• Quantum World (Entanglement):
  If the boxes are linked by quantum entanglement, they can achieve a higher combined score. Quantum mechanics predicts S can be as high as 2√2, approximately 2.828, which is greater than 2.

• The Test:
  Scientists run the experiment many times. Alice and Bob randomly choose which button to press each time. They calculate all the E(...) scores from the data and then compute S.

  • If they find S ≤ 2, the results could be explained by classical physics (pre-set instructions or slow communication)