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If you have a pure composite system whose two subsystems are in a product state, then the outcomes of measuring the subsystems (in any basis) are statistically independent. If the subsystems are entangled, then the measurement outcomes will generically be correlated.

Is there an example of an entangled state of a composite system of two isomorphic qudits, such that if you measure both subsystems in one basis, then the subsystems' measurement outcomes in that basis are independent, but if you measure both subsystems in another basis, then the outcomes are correlated?

For example, is there an entangled state $|\psi\rangle$ of two qubits such that for the joint probability mass function for a measurement in the $Z$-basis $$\left\{ P(\uparrow, \uparrow) = |\langle \uparrow \uparrow | \psi\rangle|^2, P(\uparrow, \downarrow) = |\langle \uparrow \downarrow | \psi\rangle|^2, P(\downarrow, \uparrow) = |\langle \downarrow \uparrow | \psi\rangle|^2, P(\downarrow, \downarrow) = |\langle \downarrow \downarrow | \psi\rangle|^2 \right\},$$ the measurement outcomes of the two qubits are independent, but for the joint probability mass function for a measurement of the same state in the $X$-basis $$\left\{ P(+, +) = |\langle + + | \psi\rangle|^2, P(+, -) = |\langle + - | \psi\rangle|^2,\\ P(-, +) = |\langle - + | \psi\rangle|^2, P(-, -) = |\langle - - | \psi\rangle|^2 \right\},$$ the measurement outcomes of the two qubits are dependent?

I don't see why this shouldn't be possible, but I can't think of an example.

"I don't see why this shouldn't be possible, but I can't think of an example."-- This likely has to do with complex numbers. Real examples are much trickier, if existent (for one thing, |00>+|11> is invariant under real rotations $U\otimes U$, but not under complex ones, since it is in fact invariant under $U\otimes \bar U$, so it behaves the same way under X and Z, but not under X and Y.). So using either X and Y instead of X and Z, or states with complex amplitudes as in Craig's example, is the way to go. – Norbert Schuch – 2020-08-28T10:28:22.140