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On the Cleaning Lemma of Quantum Coding Theory

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.

preprint2022arXivOpen access
Gleb KalachevSergey Sadov
math.ITInformation Theoryquant-ph
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Gleb KalachevSergey Sadov

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