= Two-spin decoherence-free encoding
{title2=$|0_L\rangle=|01\rangle,\quad|1_L\rangle=|10\rangle$}
The zero-total-$Z$ <eigenspace> of two <spin one-half> systems stores one <logical qubit>. Both basis states satisfy $(ZI+IZ)|\psi\rangle=0$, so every <collective dephasing> rotation acts as the <identity operator> on their span. Also $ZZ|01\rangle=-|01\rangle$ and $ZZ|10\rangle=-|10\rangle$, so simultaneous <Pauli Z gates> preserve every encoded <density matrix>. For independent rotations, the logical amplitudes instead acquire phases $e^{-i(\phi_1-\phi_2)/2}$ and $e^{i(\phi_1-\phi_2)/2}$, 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.
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