Reduced State Embedding for Error Correction in Quantum Cryptography

Abstract: Encoding in a high-dimensional Hilbert space improves noise resilience in quantum information processing. However, such an approach may result in cross-mode coupling and detection complexities, thereby reducing quantum cryptography performance. This fundamental trade-off between correctness and secrecy motivates the search for new error-correction approaches to better exploit the advantages of high-dimensional encoding. Here, we introduce the method of reduced state embeddings to quantum key distribution (QKD): a k-dimensional signal set embedded in a d-dimensional Hilbert space, where k<d. In the framework of quantum error correction, our reduced-state embedding realizes an explicit erasure-type error-correction within the quantum channel. We demonstrate the advantage of our scheme in realistic quantum channels, producing a higher secure key rate. We validate our approach using a d=25 QKD experimental data, derive closed-form expressions for the key rate and threshold, and determine the optimal key rate at k=5. These findings advance high-dimensional QKD and pave the way to error correction and modulation for quantum cryptography.