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arXiv:2608.05432

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

Shaikha S. Al-Qahtani, Siyao Li, Joseph J. Boutros

cs.ITquant-phSubmitted Aug 5, 2026• Announced Aug 7, 2026• 0 citations
arXivHTMLPDF

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.

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

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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.
Quantum key distribution (QKD) is a cryptographic communication protocol that utilizes quantum mechanical properties for provable absolute security against an eavesdropper. The communication is carried between two terminals using random photon polarization states represented through quantum states. Both these terminals are interconnected through disjoint quantum and classical channels. Information reconciliation using delay controlled joint decoding is performed at the receiving terminal. Its performance is characterized using data and error rates. Achieving low error rates is particularly challenging for schemes based on error correcting codes with short code lengths. This article addresses the decoding process using ordered statistics decoding for information reconciliation of both short and medium length Bose–Chaudhuri–Hocquenghem codes over a QKD link. The link’s quantum channel is modeled as a binary symmetric quantum depolarization channel, whereas the classical channel is configured with additive white Gaussian noise. Our results demonstrate the achievement of low bit error rates, and reduced decoding complexity when compared to other capacity achieving codes of similar length and configuration.
In time entanglement-based quantum key distribution (TE-QKD), Alice and Bob extract the raw key bits from the arrival times of entangled photon pairs. Each entangled pair can contribute to multiple key bits depending on how precisely Alice and Bob can measure the photon arrival times. Thus, TE-QKD can potentially increase the secret key rate compared to typical QKD implementations, which extract up to a bit per photon. Because of entanglement, the times of photon arrivals at Alice’s and Bob’s detectors and, thus, their raw keys should be identical. However, practical photon detectors suffer from time jitter errors. These errors cause discrepancies between Alice’s and Bob’s raw keys. Therefore, Alice must send information to Bob through the public channel to reconcile their raw keys. The amount of data sent for reconciliation represents a loss, rendering secret keys shorter than the raw keys. We compute the secret key rates possible in systems with detector jitter errors and show that they are much higher than those achievable in polarization entanglement-based QKD. We then construct codes for information reconciliation to approach these rates. We demonstrate that short and moderate-length standard error-correcting codes represent excellent information reconciliation choices, making TE-QKD a promising technology.
This thesis deals with soft-information based information reconciliation and data sifting for Quantum Key Distribution (QKD). A novel composite channel model for QKD is identified, which includes both a hard output quantum channel and a soft output classic channel. The Log-Likelihood Ratios, - also called soft-metrics - derived from the two channels are jointly processed at the receiver, exploiting capacity achieving soft-metric based iteratively decoded block codes. The performance of the proposed mixed-soft-metric algorithms are studied via simulations as a function of the system parameters. The core ideas of the thesis are employing Forward Error Correction (FEC) coding as opposed to two-way communication for information reconciliation in QKD schemes, exploiting all the available information for data processing at the receiver including information available from the quantum channel, since optimized use of this information can lead to significant performance improvement, and providing a security versus secret-key rate trade-off to the end-user within the context of QKD systems.
Abstract In Quantum Key Distribution (QKD), two users extract a shared secret key using a quantum communication channel in the presence of an eavesdropper. Among QKD protocols, the ones based on Energy-Time (ET) entanglement of photons have been studied extensively due to their ability to generate high key rates from the arrival times of entangled photons. For the information reconciliation (IR) stage of ET-QKD protocols (where the users communicate using a classical channel in order to reconcile differences in their data), a scheme called multi-level coding (MLC) was proposed by Zhou et al. in prior work. The MLC scheme splits the raw key symbols into bit layers and utilizes binary Low-Density Parity-Check (LDPC) codes to encode each layer. Although binary LDPC codes are able to offer low complexity decoding for IR, they have poor error-correcting performance compared to their non-binary counterparts, thus leading to low key rates. Additionally, existing LDPC codes do not fully utilize the properties of the QKD channel to optimize the key rates. In this paper, we mitigate the above issues by proposing a flexible protocol for IR in ET-QKD systems called Non-Binary Multi-Level Coding NB-MLC(a) which is parameterized by a positive integer a. The NB-MLC(a) protocol is a generalization of the MLC scheme and utilizes NB-LDPC codes from a Galois field of size 2^a. We show that by using a small value of a, the NB-MLC(a) protocol significantly improves the key rate without much increase in complexity. To further improve the key rates of the NB-MLC(a) protocol, we propose i) a joint rate and degree distribution optimization (JRDO) algorithm to design the NB-LDPC codes for the protocol; ii) an interleaved decoding and communication (IDC) scheme to decode the different layers of the NB-MLC(a) protocol. The JRDO algorithm is designed to use the QKD channel information and we show that it results in a higher key rate than codes used in prior work. Additionally, the IDC scheme improves the key rate compared to the decoding and communication methods utilized previously in literature. Overall, the NB-MLC(a) protocol that uses JRDO LDPC codes and the IDC scheme results in a significant 40 − 60% improvement in key rates compared to prior work for ET-QKD systems.
This paper deals with soft-information based information reconciliation and data sifting for Quantum Key Distribution (QKD).We propose a novel composite channel model for QKD, which includes both a hard output quantum channel and a soft output classic channel. The metrics derived from the two channels are jointly processed at the receiver, exploiting capacity achieving soft-metric based iteratively decoded block codes. The performance of the proposed mixed-softmetric algorithms are studied via simulations as a function of the system parameters. The core ideas of the paper are: a) employing FEC coding as opposed to two-way communication for information reconciliation, b) exploiting all the available information for data processing at the receiver including information available from the quantum channel, since optimized use of this information can lead to significant performance improvement.
Quantum Key Distribution (QKD) enables two parties to establish a common secret key that is information-theoretically secure by transmitting random bits that are encoded as qubits and sent over a quantum channel, followed by classical information processing steps known as information reconciliation and key extraction. Transmission of information over a quantum channel introduces errors that are generally considered to be due to the adversary's tempering with the quantum channel and needs to be corrected using classical communication over an (authenticated) public channel. Commonly used error-correcting codes in the context of QKD include cascade codes, low-density parity check (LDPC) codes, and more recently polar codes. In this work, we explore the applicability of designing of a polar code encoder based on a channel reliability sequence. We show that the reliability sequence can be derived and used to design an encoder independent of the choice of decoder. We then implement our design and evaluate its performance against previous implementations of polar code encoders for QKD as well as other typical error-correcting codes. A key advantage of our approach is the modular design which decouples the encoder and decoder design and allows independent optimization of each. Our work leads to more versatile polar code-based error reconciliation in QKD systems that would result in deployment in a broader range of scenarios.