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arXiv:1011.1190 · Physical Review A

Min-entropy and quantum key distribution: Nonzero key rates for “small” numbers of signals

Sylvia Bratzik, Markus Mertz, Hermann Kampermann, Dagmar Bruß

Published Feb 28, 2011• 26 citations
arXivHTMLPDF

Abstract

We calculate an achievable secret key rate for quantum key distribution with a finite number of signals by evaluating the quantum conditional min-entropy explicitly. The min-entropy for a classical random variable is the negative logarithm of the maximal value in its probability distribution. The quantum conditional min-entropy can be expressed in terms of the guessing probability, which we calculate for $d$-dimensional systems. We compare these key rates to previous approaches using the von Neumann entropy and find nonzero key rates for a smaller number of signals. Furthermore, we improve the secret key rates by modifying the parameter estimation step. Both improvements taken together lead to nonzero key rates for only ${10}^{4}$--${10}^{5}$ signals. An interesting conclusion can also be drawn from the additivity of the min-entropy and its relation to the guessing probability: for a set of symmetric tensor product states, the optimal minimum-error discrimination (MED) measurement is the optimal MED measurement on each subsystem.

DOI
10.1103/physreva.83.022330

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We calculate an achievable secret key rate for quantum key distribution with a finite number of signals by evaluating the quantum conditional min-entropy explicitly. The min-entropy for a classical random variable is the negative logarithm of the maximal value in its probability distribution. The quantum conditional min-entropy can be expressed in terms of the guessing probability, which we calculate for ddd-dimensional systems. We compare these key rates to previous approaches using the von Neumann entropy and find nonzero key rates for a smaller number of signals. Furthermore, we improve the secret key rates by modifying the parameter estimation step. Both improvements taken together lead to nonzero key rates for only 104{10}^{4}104--105{10}^{5}105 signals. An interesting conclusion can also be drawn from the additivity of the min-entropy and its relation to the guessing probability: for a set of symmetric tensor product states, the optimal minimum-error discrimination (MED) measurement is the optimal MED measurement on each subsystem.
Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.
In quantum Shannon theory, various kinds of quantum entropies are used to characterize the capacities of noisy physical systems. Among them, min-entropy and its smooth version attract wide interest especially in the field of quantum cryptography as they can be used to bound the information obtained by an adversary. However, calculating the exact value or nontrivial bounds of min-entropy are extremely difficult because the composite system dimension may scale exponentially with the dimension of its subsystem. Here, we develop a one-shot lower bound calculation technique for the min-entropy of a classical-quantum state that is applicable to both finite and infinite dimensional reduced quantum states. Moreover, we show our technique is of practical interest in at least three situations. First, it offers an alternative tight finite-data analysis for the BB84 quantum key distribution scheme. Second, it gives the best finite-key bound known to date for a variant of device independent quantum key distribution protocol. Third, it provides a security proof for a novel source-independent continuous-variable quantum random number generation protocol. These results show the effectiveness and wide applicability of our approach.
Abstract Quantum key distribution (QKD) allows for secure communications safe against attacks by quantum computers. QKD protocols are performed by sending a sizeable, but finite, number of quantum signals between the distant parties involved. Many QKD experiments, however, predict their achievable key rates using asymptotic formulas, which assume the transmission of an infinite number of signals, partly because QKD proofs with finite transmissions (and finite-key lengths) can be difficult. Here we develop a robust numerical approach for calculating the key rates for QKD protocols in the finite-key regime in terms of two semi-definite programs (SDPs). The first uses the relation between conditional smooth min-entropy and quantum relative entropy through the quantum asymptotic equipartition property, and the second uses the relation between the smooth min-entropy and quantum fidelity. The numerical programs are formulated under the assumption of collective attacks from the eavesdropper and can be promoted to withstand coherent attacks using the postselection technique. We then solve these SDPs using convex optimization solvers and obtain numerical calculations of finite-key rates for several protocols difficult to analyze analytically, such as BB84 with unequal detector efficiencies, B92, and twin-field QKD. Our numerical approach democratizes the composable security proofs for QKD protocols where the derived keys can be used as an input to another cryptosystem.
We compare three proof techniques for composable finite-size security of quantum key distribution under collective attacks, with emphasis on how the resulting secret-key rates behave at practically relevant block lengths. As a benchmark, we consider the BB84 protocol and evaluate finite-size key-rate estimates obtained from entropic uncertainty relations (EUR), from the asymptotic equipartition property (AEP), and from a direct finite-block analysis based on the conditional min-entropy, which we refer to as the finite-size min-entropy (FME) approach. For BB84 we show that the EUR-based bound provides the most favorable performance across the considered parameter range, while the AEP bound is asymptotically tight but can become overly pessimistic at moderate and small block sizes, where it may fail to certify a positive key. The FME approach remains effective in this small-block regime, yielding nonzero rates in situations where the AEP estimate vanishes, although it is not asymptotically optimal for BB84. These results motivate the use of FME-type analyses for continuous-variable protocols in settings where tight EUR-based bounds are unavailable, notably for coherent-state schemes where current finite-size analyses typically rely on AEP-style corrections.
Quantum key distribution (QKD) is a form of secure communication which transmits secret keys between two trusted parties. Evaluation of the achievable secret key rates are essential to ensure robust security of any QKD protocols. As such, the optimization of the secret key rate to obtain tight and reliable bounds are required especially for instances when the number of transmitted signals are finite. In this work, we make use of the generalized Rényi entropic quantities to obtain a tighter bound on the key rate compared to previous works which uses the von Neumann entropy. However, the optimization of Rényi entropic quantities is often non-trivial. Hence, we derive a generalized bound on the Rényi entropy in terms of the Rényi divergence which allows us to apply the Frank-Wolfe minimization algorithm for the optimization of secret key rates, thus generalizing existing numerical frameworks. With this generalized framework, we show improvements in regimes where finite signals experience high loss, thus ensuring relevance for real-world scenarios, particularly for satellite-based QKD implementations.