= Exchange symmetry obstruction to individual spin control
{title2=$[H(t),S]=0\Rightarrow[U(t),S]=0$}
Let $S$ be the <SWAP gate>. If the drift <Hamiltonian operator> and every allowed control <Hamiltonian operator> commute with $S$, every reachable <time-evolution operator> also commutes with $S$. Indeed, $SUS$ satisfies the same initial-value <Schrödinger equation> as $U$ and starts at $I$, so uniqueness gives $SUS=U$. Yet $S(Z\otimes I)S=I\otimes Z\ne Z\otimes I$. An individual <Pauli Z gate> is consequently unavailable within that symmetric control model. Equal nuclear resonances, isotropic <exchange interaction>, and identical coupling to a common homogeneous <radio frequency> field provide an example. An added interaction that breaks this symmetry can remove the obstruction.
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