Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators

Abstract We study bipartite unitary operators which stay invariant under the local actions of diagonal and orthogonal groups. investigate structural properties these operators, arguing that symmetry makes them suitable for analytical study. As a first application, we construct large new families dual gates in arbitrary finite dimensions, are important toy models entanglement spreading quantum circuits. then analyze non-local nature both discrete (operator Schmidt rank) continuous (entangling power) settings. Our scrutiny reveals can be used to simulate any gate via stochastic operations classical communication. Furthermore, establish one-to-one connection between set with maximum entangling power complex Hadamard matrices. Finally, discuss distinguishability setting stated symmetry.