 Two-spin decoherence-free encoding

The zero-total-$Z$ eigenspace of two spin one-half systems stores one logical qubit. Both basis states satisfy $(ZI+IZ)|\psi ⟩=0$, so every collective dephasing rotation acts as the identity operator on their span. Also $ZZ|01⟩=-|01⟩$ and $ZZ|10⟩=-|10⟩$, so simultaneous Pauli Z gates preserve every encoded density matrix. For independent rotations, the logical amplitudes instead acquire phases $e^{-i({\varphi }_1-{\varphi }_2)/2}$ and $e^{i({\varphi }_1-{\varphi }_2)/2}$, which generally alter their relative phase. The code is not a two-dimensional decoherence-free subspace for arbitrary collective rotations about all axes; the invariant two-spin singlet alone is only one-dimensional.