On generating r-uniform subspaces with the isometric mapping method

We propose a compositional approach to construct subspaces consisting entirely of r-uniform states, including the ones in heterogeneous systems. The approach allows one to construct new objects from old ones: it combines encoding isometries of pure quantum error correcting codes with entangled multipartite states and subspaces. The presented methods can be also used to construct new pure quantum error correcting codes from certain combinations of old ones. The approach is illustrated with various examples including constructions of 2-, 3-, 4-, 5-uniform subspaces. The results are then compared with analogous constructions obtained with the use of orthogonal arrays.