= Heisenberg exchange echo
{c}
{title2=$PUPU=e^{-iJtZ\otimes Z}$}
For $H=J(XX+YY+ZZ)/2$, $U=e^{-itH}$, and an individually applied <Pauli Z gate> $P=Z\otimes I$, conjugation gives $PHP=J(-XX-YY+ZZ)/2$. The paired <Pauli operators> $XX,YY,ZZ$ commute with one another, so
$$
PUPU=e^{-itPHP}e^{-itH}=e^{-iJtZZ}.
$$
Two equal free-evolution intervals, each of duration $t$, therefore cancel exchange while retaining the <Ising coupling>. This <spin echo> is exact for instantaneous ideal gates; a finite pulse requires accounting for the interaction during the pulse. A simultaneous <Pauli Z gate> on each spin commutes with $H$ and cannot perform the same cancellation. The physical availability of the individual gate is a separate <quantum control> requirement.
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