SWAP and Transpose by displacements, Stabilizer Renyi entropies for continuous variables and qudits and other applications

Abstract

In this note, I highlight a useful formula for the SWAP operator as an average of anti-correlated Heisenberg-Weyl displacements, valid for arbitrary-dimensional quantum systems. As an application I show how the relation can be used to quickly prove normalization identities for the Weyl function, and apply the result to Weyl magic and Wigner magic as the generalization of the recently suggested Renyi Stabilizer entropy to q-dits and CV.