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

TL;DR

This paper presents a formula for the SWAP operator using displacements in quantum systems, enabling new proofs and applications in quantum information theory, including generalizations of stabilizer and magic measures for qudits and continuous variables.

Contribution

It introduces a novel formula for the SWAP operator as an average of displacements, facilitating proofs and extending stabilizer and magic entropy concepts to higher-dimensional and continuous-variable systems.

Findings

Derived a formula for SWAP as an average of displacements

Applied the formula to prove normalization identities for Weyl functions

Extended stabilizer and magic entropy measures to qudits and continuous variables

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.