Entanglement-assisted quantum error correcting codes over nice rings
Sang-Jun Lee, Andreas Klappenecker · 2014
Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.