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)


It is the tensor product of the Pauli X with itself. Preskill is specifying the kind of correlation by giving you a matrix with a +1 eigenspace corresponding to the desired set of states. States where the simultaneous application of an $X$ gate to each qubit has no effect, including phase kickback when conditioned on an ancilla qubit.

This notation is very common in the stabilizer formalism. For example, error correcting codes that are stabilizer codes are often specified by a set of tensor-products-of-Paulis that must be repeatedly measured, e.g. $X_1 X_3 Z_5 Z_7$. When errors occur, the state will flip between the +1 and -1 eigenspaces of some of the measured stabilizers.


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