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arXiv:2603.18213

Preprocessing noise in finite-size quantum key distribution

Gabriele Staffieri, Giuseppe D'Ambruoso, Giovanni Scala, Cosmo Lupo

quant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations• Revised Oct 30, 2026 (v2)
v1 · Mar 18, 2026v2 · Oct 30, 2026
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Abstract

It is known that trusted, preprocessing noise may boost quantum key distribution by expanding the range of values of tolerated noise. For BB84, adding trusted noise may allow the generation of secret keys even for qubit error rate (QBER) beyond the 11% threshold in the asymptotic regime. Here we study the effect of preprocessing noise in the finite-size regime where only a limited number of signals are exchanged between Alice and Bob. We compute the secret key rate from the full variational formulation of the sandwiched Renyi conditional entropy, optimized through nonsymmetric conic optimization. We find that trusted noise improves the key rate only for a finite interval of alpha, from the alpha approaching 1 limit up to alpha approximately 1.4. We then optimize the finite-size key rate jointly over both q, alpha, and the fraction of raw keys used for parameter estimation, to study how trusted preprocessing changes the maximum tolerable QBER as the block size increases. We show that the secret-key rate enhancement provided by optimized noisy preprocessing becomes more pronounced at higher QBER levels and can increase the maximum tolerable QBER over a broad range of finite block sizes.

Comments
14 pages, 9 figures

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Quantum Key Distribution (QKD) allows two legitimate parties, Alice and Bob, to share a secret key with information-theoretic security, ensuring that an eavesdropper, Eve, cannot obtain any information about the key. A QKD protocol is provably secure only when the Quantum Bit Error Rate (QBER) is below a certain threshold and, therefore, the presence of noise inherently limits the maximum transmission distance. Advantage Distillation (AD) is a classical post-processing technique that enhances QKD protocols by increasing the maximum acceptable QBER and, thus, allows extending the communication range. AD operates by post-selecting blocks of bits and extracting fewer but highly correlated bits between Alice and Bob, which exhibit a reduced QBER, thus lowering the amount of information that has to be disclosed during the information reconciliation step. In this study, we present a new analytical expression for the secure key rate. This will enable a finite-size secure key length analysis of the decoy state version of the BB84 protocol including AD post-processing.
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[[abstract]]An enhanced version of the quantum key distribution (QKD) protocol in quantum cryptography is proposed. We consider that realistic detectors and imperfect devices may introduce noise into the transmission thereby causing a high error rate. The proposed scheme is based on an error rate analysis and a random sampling procedure in statistics to produce a probabilistic bound on the error estimation. Confidence intervals are used to interpret the error estimate with a specified confidence level. Based on the information obtained, Alice and Bob judge what caused the errors and establish confidence levels for the error rate of the remaining untested bits. Hence, they can determine the maximum number of times reconciliation needs to be performed to remove all errors from what remains of a raw key to produce an error-free secret key. As a result, the tedious reconciliation process can be reduced and a neglectful probability of not detecting the existence of remaining errors can be avoided as well. The privacy amplification is then applied: (1) to extract a secret key from a partially secret key, and (2) to enlarge the length of the secret key into the final key. The proposed scheme can work when dealing with errors caused by random noise and eavesdropping.