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As far as I know, correlation of two qubits in the $X$ basis implies that under a simultaneous bit flip, the composite quantum state must be invariant. For instance, $A$ and $B$ can be said to be correlated in the $X$ basis if they share a Bell state i.e.

$$|\Psi\rangle_{AB} = \frac{1}{\sqrt{2}}(|0\rangle_A\otimes |0\rangle + |1\rangle_A\otimes |1\rangle_B).$$

Under simultaneous bit flip it becomes

$$|\Psi\rangle_{AB}' = \frac{1}{\sqrt{2}}(|1\rangle_A\otimes |1\rangle_B + |0\rangle_A\otimes |0\rangle_B).$$

Clearly, $$|\Psi\rangle_{AB}=|\Psi\rangle_{AB}'.$$

Preskill uses the notation $X\otimes X = 1$ to denote this kind of correlation ^{[1]}. He also uses a similar notation $Z\otimes Z$ to imply correlation in the $Z$ basis. Any idea where this notation comes from? It doesn't seem obvious to me. At first sight, I would have considered $X\otimes X$ to mean the tensor product of the Pauli-$X$ operator with itself.

^{[1]: John Preskill - Introduction to Quantum Information (Part 1) - CSSQI 2012 (~44:10)}