Classification of Schmidt-rank-two multipartite unitary gates by singular number

The multipartite unitary gates are called genuine if they not product operators across any bipartition. We mainly investigate the classification of Schmidt rank two, by focusing on multiqubit scenario. For (excluding bipartite) there is an essential fact that their decompositions unique. Based this fact, we propose a key notion named as singular number to classify concerned. defined local in decomposition. then determine accurate range number. each number, formulate parametric under equivalence. Finally, extend study three-qubit diagonal due close relation between and Schmidt-rank-two unitaries. start with discussing two typical examples one which fundamental gate, i.e., CCZ gate. Then characterize greater than two. show gate has at most three, present necessary sufficient condition for such three. This completes characterization all gates.