Reduced-state embedding for quantum cryptography

Encoding in a high-dimensional Hilbert space improves noise resilience in quantum information processing. This approach, however, 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 quantum error-correction approaches for cryptography. Here, we introduce state embeddings that use a k-symbol subset within a d-dimensional Hilbert space, tailored to the channel's error structure. 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 quantum key distribution (QKD) experimental data, derive closed-form expressions for the key rate and threshold, and determine the optimum at k=5. These findings advance high-dimensional QKD and pave the way to error-correction and modulation for quantum cryptography.