Quantum error-correcting codes via inner products and error bases

TL;DR

This paper introduces a novel framework for quantum error correction using inner products and error bases, providing new conditions for code existence that deepen the theoretical understanding of quantum information protection.

Contribution

It develops a self-consistent theory based on inner products and error bases, extending quantum error correction foundations beyond classical analogies.

Findings

New necessary and sufficient conditions for quantum error-correcting codes

Extension of quantum error correction theory beyond classical analogies

Structural insights from operator theory and product spaces

Abstract

In this paper, we address the problem of state communication in finite-level quantum systems through noise-affected channels. Our approach is based on a self-consistent theory of decoding inner products associated with the code and error (or noise) bases defined on corrupting subspaces. This viewpoint yields new necessary and sufficient conditions for the existence of quantum error-correcting codes in terms of these inner products. The obtained results extend the foundations of quantum error correction beyond classical analogies, highlighting the structural insights offered by operator theory and the underlying product space.