Alice and Bob play a cooperative game against a referee. They can agree on a strategy beforehand but cannot communicate during play. No classical strategy wins more than 75% of the time. With shared entangled qubits, they reach cos²(π/8) = 85.36%. This is the experimental signature that nature is not classical — Clauser/Aspect/Zeilinger won the 2022 Nobel Prize for proving it.
x ∈ {0, 1} and Bob a bit y ∈ {0, 1}, chosen uniformly at random. Alice outputs a ∈ {0, 1}; Bob outputs b ∈ {0, 1}. They win if a XOR b = x AND y. So: if either x or y is 0, they need equal outputs. If both x and y are 1, they need opposite outputs.
75%. Proof: the four win conditions are a=b (when xy=00), a=b (when xy=01), a=b (when xy=10), a≠b (when xy=11). If we sum the four "agreement" values (treating outputs as ±1: ⟨A_xB_y⟩), classical strategies must satisfy ⟨A_0B_0⟩ + ⟨A_0B_1⟩ + ⟨A_1B_0⟩ − ⟨A_1B_1⟩ ≤ 2 — that's the CHSH inequality.
|ψ⟩ = (|01⟩ − |10⟩)/√2. Alice measures along axis at angle 0 (if x=0) or 45° (if x=1). Bob measures along 22.5° (if y=0) or −22.5° (if y=1). The quantum prediction for the agreement is ⟨A_xB_y⟩ = −cos(θ_A − θ_B). Plug in the angles: each of the four terms equals ±1/√2, giving S = 2√2 ≈ 2.828 — beating the classical bound by 41%.