Diversity in Coded TE-QKD Channels: Achieving Infinite Diversity out of Finite System Resources

Abstract:We establish conditions and give proofs on how an error-correcting code can attain infinite diversity in a time-entanglement quantum key distribution (TE-QKD) reconciliation. The shocking result, never encountered in the literature on coding and communication theory, is that a decoder exhibits an infinite diversity order while the channel has finite diversity and the code has a relatively short finite length. This paper studies the diversity order of coded TE-QKD reconciliation, defined by the asymptotic slope of the error probability at high signal-to-noise ratio. For bounded-distance algebraic decoding, we derive a necessary and sufficient condition in terms of the number of photons per codeword and the decoding radius. For soft-decision decoding, we introduce the maximal finite diversity (MFD) property and prove that infinite diversity is achieved if and only if the code is MFD deficient. The infinite diversity in TE-QKD has no counterpart in classical fading channels, where decoding can only multiply a finite diversity order by a finite factor. Examples of short codes based on Golay, Reed-Solomon, Bose-Chaudhuri-Hocquenghem (BCH), and Reed-Muller codes validate the analysis and illustrate how the TE-QKD system parameters and the relatively short code parameters affect the achievable diversity for both hard and soft information reconciliation.

The manuscript has 54 pages, 16 figures, and 8 tables. We intend to submit this manuscript to an IEEE journal

Information Theory (cs.IT); Quantum Physics (quant-ph)

arXiv:2608.05432 [cs.IT]

(or arXiv:2608.05432v1 [cs.IT] for this version)

https://doi.org/10.48550/arXiv.2608.05432 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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