 Solution

Quantum error correction encodes a logical qubit redundantly, obtains an error syndrome without measuring its unknown logical amplitudes, and applies a recovery that reverses the specified errors. A decoherence-free subspace instead uses a symmetry of a specified noise model: each noise operator acts as a scalar on the entire encoded space, so no syndrome-dependent recovery is required. This is passive protection and can also be viewed as a special case of quantum error correction.

For the two-spin decoherence-free encoding, chooseSimultaneous Pauli Z gates givesoBoth logical components acquire the same global phase; their relative phase and amplitudes survive. The density matrix is exactly unchanged, so even an unknown random choice between applying $I$ and $Z\otimes Z$ causes no logical dephasing.

More strongly, the encoding protects against arbitrary collective dephasing:because both basis states have total $Z$ eigenvalue zero. At $\varphi =\pi$, $R_z(\pi )\otimes R_z(\pi )=-Z\otimes Z$, consistent with the previous overall phase. The protection relies on correlated noise: independent $Z$ rotations generally change the logical relative phase, and this two-spin code does not protect a whole logical qubit against arbitrary collective rotations about all three axes or against arbitrary single-spin errors.