Singular value decomposition and matrix reorderings in quantum information theory

TL;DR

This paper reviews matrix decompositions and reorderings in quantum information theory, providing a unified notation and simple formulas for quantum channels, supported by Mathematica implementations.

Contribution

It introduces a unified notation for matrix decompositions and reorderings, simplifying the analysis of quantum states and operations in quantum information theory.

Findings

Derived simple formulas for quantum channel composition

Provided Mathematica package for quantum information calculations

Clarified the correspondence between quantum states and operations

Abstract

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyse the correspondence between quantum states and operations with the help of Jamiolkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.