= Product-state obstruction to SWAP universality
{title2=$\mathrm{SWAP}(A\otimes B)=(B\otimes A)\mathrm{SWAP}$}
The <SWAP gate> and local <unitary operators> preserve <product states>. The displayed identity moves every exchange past a pair of local gates, reducing any such two-<qubit> <quantum circuit> to a local operation, optionally followed by <SWAP gate>. On more wires the same argument leaves a permutation of the tensor factors. By induction, a fully factored input remains factored. Since <universal quantum computation> includes operations that create <quantum entanglement> from a <product state>, these gates cannot form a <universal quantum gate set>.
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