On the Cleaning Lemma of Quantum Coding Theory

TL;DR

This paper reveals that the core of the Cleaning Lemma in Quantum Coding Theory is fundamentally a linear algebra fact, which can be simplified to a combinatorial lattice-theoretical level, leading to new variants and propositions.

Contribution

It provides a simplified, unified mathematical foundation for the Cleaning Lemma, connecting linear algebra and combinatorics in quantum coding theory.

Findings

Core of the Cleaning Lemma is a simple linear algebra fact

Reduction to a combinatorial, lattice-theoretical level

Derivation of new variants and propositions

Abstract

The term "Cleaning Lemma" refers to a family of similar propositions that have been used in Quantum Coding Theory to estimate the minimum distance of a code in terms of its length and dimension. We show that the mathematical core is a simple fact of linear algebra of inner product spaces; moreover, it admits a further reduction to a combinatorial, lattice-theoretical level. Several concrete variants of the Cleaning Lemma and some additional propositions are derived as corollaries of the proposed approach.