Topological sum rule for geometric phases of quantum gates

We establish a topological sum rule, ${\nu }_U=\frac{1}{2\pi }{\sum }_n{\gamma }_n=m{\nu }_H$, connecting the geometric phases accumulated by a two-qubit system over a complete basis of initial states to the winding number ${\nu }_H$ classifying its Hamiltonian. Implementations of the same gate from different topological classes must distribute these phases differently, making their distinction measurable through the Wootters concurrence. As a corollary, nontrivial topology is a necessary condition for entanglement: only Hamiltonians with access to ${\nu }_H\mathrm{\ne }0$ can generate it.