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This question is taken from https://github.com/quantumlib/Cirq/issues/3032.

The "standard" protocol measures Alice's and Bob's qubits in different bases, while the one in Cirq measures in the same basis. I can't prove the equivalence of these seemly different protocols that violate Bell's inequality.

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The part you're overlooking is these lines:

    # Players do a sqrt(X) based on their referee's coin.
    circuit.append([
        cirq.CNOT(alice_referee, alice)**0.5,
        cirq.CNOT(bob_referee, bob)**0.5,
    ])

The players rotate their qubits conditioned on what the local referee said. This is equivalent to changing the measurement basis conditioned on what the local referee said.

The only relevant property is that the effective measurement bases start off offset by -45 degrees (mostly agree), so that a 90 degree rotation by one player gets you to +45 degrees (still mostly agree) while a rotation by both players gets you to +135 degrees (mostly disagree). The example sets up such a situation.

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  • $\begingroup$ Thanks for clearing this up. Can you add a section to your answer about how the measurement basis don't matter, they just have to be rotated a certain way relative to each other? $\endgroup$ – Victory Omole Aug 20 '20 at 2:22

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