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arXiv:1112.5542 · Physical Review A

Quantum key distribution with finite resources: Taking advantage of quantum noise

Markus Mertz, Hermann Kampermann, Z. Shadman, Dagmar Bruß

Published Apr 9, 2013• 13 citations
arXivHTMLPDF

Abstract

We compare the effect of different noise scenarios on the achievable rate of an ${\ensuremath{\varepsilon}}_{}$-secure key for the BB84 and the six-state protocol. We study the situation where quantum noise is added deliberately, and investigate the remarkable benefit for the finite key rate. We compare our results to the known case of added classical noise and the asymptotic key rate, i.e., in the limit of infinitely many signals. As a complementary interpretation, we show that under the realistic assumption that the noise which is unavoidably introduced by a real channel is not fully dedicated to the eavesdropper, the secret key rate increases significantly.

DOI
10.1103/physreva.87.042312

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We compare the effect of different noise scenarios on the achievable rate of an \ensuremathε{\ensuremath{\varepsilon}}_{}\ensuremathε​-secure key for the BB84 and the six-state protocol. We study the situation where quantum noise is added deliberately, and investigate the remarkable benefit for the finite key rate. We compare our results to the known case of added classical noise and the asymptotic key rate, i.e., in the limit of infinitely many signals. As a complementary interpretation, we show that under the realistic assumption that the noise which is unavoidably introduced by a real channel is not fully dedicated to the eavesdropper, the secret key rate increases significantly.
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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.
We study eavesdropping in quantum key distribution with the six state protocol, when the signal states are mixed with white noise. This situation may arise either when Alice deliberately adds noise to the signal states before they leave her lab, or in a realistic scenario where Eve cannot replace the noisy quantum channel by a noiseless one. We find Eve's optimal mutual information with Alice, for individual attacks, as a function of the qubit error rate. Our result is that added quantum noise reduces Eve's mutual information more than Bob's.
Quantum key distribution protocols have been studied for almost four decades, and have already seen real world applications. Such implementations often operate in naturally noisy environments (e.g., sensor thermal noise, sun radiation, etc.) with known or measurable noise levels. However, existing security proofs implicitly attribute any channel noise to a possible eavesdropper, an overly pessimistic assumption that can adversely effect the guaranteed protocol key-rate. In this paper, we analyze general prepare-and-measure QKD schemes, and show how to improve the bounds on the key-rate under the assumption of an independent state replacement noise at the receiver. Taking BB84 as a concrete example, our results theoretically guarantee a higher key-rate, as a function of the minimal possible replacement probability. In practice, for some parameter-regimes, this guarantees a higher tolerable QBER.