Matrix reduction and Lagrangian submodules

TL;DR

This paper introduces algorithms for diagonalizing submodules of Z_d^n, describes Lagrangian submodules in symplectic bases, and defines a concept called fringe for Gram matrices to identify nearly symplectic submodules, advancing geometric tools in quantum information.

Contribution

It provides new algorithms for diagonalization and classification of submodules in symplectic geometry relevant to quantum information theory.

Findings

Algorithm for diagonal basis matrices of Z_d-modules

Explicit description of Lagrangian submodules in symplectic bases

Introduction of fringe concept for Gram matrices

Abstract

This paper deals with three technical ingredients of geometry for quantum information. Firstly, we give an algorithm to obtain diagonal basis matrices for submodules of the Z_{d}-module Z_{d}^{n} and we describe the suitable computational basis. This algorithm is set along with the mathematical properties and tools that are needed for symplectic diagonalisation. Secondly, with only symplectic computational bases allowed, we get an explicit description of the Lagrangian submodules of Z_{d}^{2n}. Thirdly, we introduce the notion of a fringe of a Gram matrix and provide an explicit algorithm using it in order to obtain a diagonal basis matrix with respect to a symplectic computational basis whenever possible. If it is possible, we call the corresponding submodule nearly symplectic. We also give an algebraic property in order to single out symplectic submodules from nearly symplectic ones.