= Solution
<Quantum error correction> detects an error syndrome and actively restores a logical state within a correctable error set. A <decoherence-free subspace> instead encodes into a subspace on which every allowed noise operator acts as the same scalar; no relative logical phase or mixing is produced by that noise. It relies on the specified correlation structure, and does not protect against arbitrary independent errors.
For collective rotations $R_z(\phi)\otimes R_z(\phi)$,
$$
|01\rangle\mapsto e^{-i\phi/2}e^{i\phi/2}|01\rangle=|01\rangle,
\qquad |10\rangle\mapsto|10\rangle.
$$
Thus every encoded superposition $\alpha|01\rangle+\beta|10\rangle$ is invariant. Equivalently it lies in the zero-eigenvalue subspace of $Z\otimes I+I\otimes Z$.
By contrast,
$$
\alpha|00\rangle+\beta|11\rangle
\mapsto\alpha e^{-i\phi}|00\rangle+\beta e^{i\phi}|11\rangle.
$$
The relative phase changes by $2\phi$, although $Z\otimes Z$ acts as $+I$ on both basis states. Invariance under one discrete operation does not establish invariance under its continuous rotation family. A concrete counterexample is $\phi=\pi/2$: the equally weighted encoded plus state becomes the encoded minus state, up to a global phase, and is orthogonal to its original state.
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