 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 $R_z(\varphi )\otimes R_z(\varphi )$,Thus every encoded superposition $\alpha |01⟩+\beta |10⟩$ is invariant. Equivalently it lies in the zero-eigenvalue subspace of $Z\otimes I+I\otimes Z$.

By contrast,The relative phase changes by $2\varphi$, 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 $\varphi =\pi /2$: the equally weighted encoded plus state becomes the encoded minus state, up to a global phase, and is orthogonal to its original state.