On generating r-uniform subspaces with the isometric mapping method

TL;DR

This paper introduces a compositional method for constructing r-uniform subspaces in quantum systems, leveraging encoding isometries and entangled states to generate new quantum error-correcting codes and subspaces.

Contribution

It presents a novel approach combining isometric mappings with entangled states to construct r-uniform subspaces and quantum error-correcting codes, including heterogeneous systems.

Findings

Constructed 2-, 3-, 4-, 5-uniform subspaces using the method.

Demonstrated the method's ability to generate new quantum error-correcting codes.

Compared the new constructions with orthogonal array-based methods.

Abstract

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.