On the logical operators of quantum codes

TL;DR

This paper introduces a symplectic Gram-Schmidt orthogonalization method to identify logical operators in quantum codes, especially useful for codes derived from classical codes and entanglement-assisted codes, providing new formulas for entanglement calculation.

Contribution

It presents a novel orthogonalization approach to determine logical operators, enhancing understanding of quantum codes from classical codes and entanglement requirements.

Findings

Provides a new method for finding logical operators in quantum codes.

Offers formulas for calculating entanglement in entanglement-assisted codes.

Improves analysis of quantum codes derived from classical codes.

Abstract

I show how applying a symplectic Gram-Schmidt orthogonalization to the normalizer of a quantum code gives a different way of determining the code's logical operators. This approach may be more natural in the setting where we produce a quantum code from classical codes because the generator matrices of the classical codes form the normalizer of the resulting quantum code. This technique is particularly useful in determining the logical operators of an entanglement-assisted code produced from two classical binary codes or from one classical quaternary code. Finally, this approach gives additional formulas for computing the amount of entanglement that an entanglement-assisted code requires.