In this talk, I will present a beautiful—yet seemingly little-known—relation that expresses the SWAP operator in quantum mechanics as an average over Heisenberg-Weyl displacements.

This relation holds for qubits, qudits, and even continuous-variable systems in quantum optics.

I will demonstrate how this result can be used to quickly establish certain normalization identities and to define “displacement-invariant” measures that are useful in quantum computation, particularly as indicators of how far a quantum state is from being a stabilizer state.

As a second application, I will show how the SWAP-by-displacement relation leads to a new method for computing entanglement entropy. In particular, I will demonstrate how the calculation works for Gaussian fields, relate it to the strong Szegő formula (including a new heuristic derivation of the formula!), and discuss its implications for a potential new approach to proving the Ryu–Takayanagi formula in AdS/CFT.