Questions- show that cali-pairs would violate local realism

Question

Correlated Pairs-

Consider a hypothetical type of correlated pairs (titled Cali-Pairs), with 3 possible measurements/inputs for each particle:

-x₀, x₁, x₂ for X

-y₀, y₁, y₂ for Y

Each input can produce two outputs: -1 or 1.

The correlations are as follows:

-For each i, X_i = Y_i (so p (X_i = Y_i |x_iy_i) = 1)

-For every i = j, X_i ≠ Y_i (so p (X_i ≠ Y_j |x_iy_i) = 1 for i ≠ j)

The sum of the probabilities (titled Sum) would then add up to:

p (X₀ = Y₀) + p (X₀ ≠ Y₁) + p (X₀ ≠ Y₂) +

P (X₁ = Y₁) + P (X_i ≠ Y₀) + P (X₁ ≠ Y₂) +

p (X₂ = Y₂) + p (X₂ ≠ Y₀) + p (X₂ ≠ Y₁) +

Worth noting though is that 0 <= Sum <= 9

Cali-pairs saturate this inequality.

Questions-

1. Show that Cali-pairs would violate local realism.

2. Find the max value of Sum for pairs that satisfy local realism.

3. Consider the entangled state for photon polarization:

|A > = 1/√2[|00 > +|11 >1]

Show that by choosing angles for x_i and y_i appropriately, quantum mechanics gives a value for Sum that exceeds the local realistic max.

Find the max quantum value of Sum.