Matrix reduction and Lagrangian submodules

O. Albouy (1)

1 IPNL - Institut de Physique Nucléaire de Lyon

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.

Mots clés: finite quantum mechanics; finitely generated modules; symplectic product; matrix reduction; Lagrangian submodules; Gram matrix; nearly symplectic submodules

Domaines: Physique Quantique [quant-ph]

O. Albouy. Matrix reduction and Lagrangian submodules. 2008. ⟨in2p3-00319021⟩