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arXiv:2105.05342 · IEEE Transactions on Information Theory

Privacy Amplification and Decoupling Without Smoothing

Frédéric Dupuis

Published Aug 28, 2023• 33 citations
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Abstract

We prove an achievability result for privacy amplification and decoupling in terms of the sandwiched Rényi entropy of order$\alpha \in (1,2]$; this extends previous results which worked for$\alpha =2$. The fact that this proof works for$\alpha $close to 1 means that we can bypass the smooth min-entropy in the many applications where the bound comes from the fully quantum AEP or entropy accumulation, and carry out the whole proof using the Rényi entropy, thereby easily obtaining an error exponent for the final task. This effectively replaces smoothing, which is a difficult high-dimensional optimization problem, by an optimization problem over a single real parameter$\alpha $.

DOI
10.1109/tit.2023.3301812

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We prove an achievability result for privacy amplification and decoupling in terms of the sandwiched Rényi entropy of orderα∈(1,2]\alpha \in (1,2]α∈(1,2]; this extends previous results which worked forα=2\alpha =2α=2. The fact that this proof works forα\alpha αclose to 1 means that we can bypass the smooth min-entropy in the many applications where the bound comes from the fully quantum AEP or entropy accumulation, and carry out the whole proof using the Rényi entropy, thereby easily obtaining an error exponent for the final task. This effectively replaces smoothing, which is a difficult high-dimensional optimization problem, by an optimization problem over a single real parameterα\alpha α.
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.
Quantum privacy amplification is a central task in quantum cryptography. Given shared randomness, which is initially correlated with a quantum system held by an eavesdropper, the goal is to extract uniform randomness which is decoupled from the latter. The optimal rate for this task is known to satisfy the strong converse property and we provide a lower bound on the corresponding strong converse exponent. In the strong converse region, the distance of the final state of the protocol from the desired decoupled state converges exponentially fast to its maximal value, in the asymptotic limit. We show that this necessarily leads to totally insecure communication by establishing that the eavesdropper can infer any sent messages with certainty, when given very limited extra information. In fact, we prove that in the strong converse region, the eavesdropper has an exponential advantage in inferring the sent message correctly, compared to the achievability region. Additionally we establish the following technical result, which is central to our proofs, and is of independent interest: the smoothing parameter for the smoothed max-relative entropy satisfies the strong converse property.
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[12,∞)α\in[\frac{1}{2},\infty)α∈[21​,∞). For α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞)α\in[1,\infty)α∈[1,∞), it is characterized by the order-ααα sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for α∈(2,∞)α\in(2,\infty)α∈(2,∞), expressed in terms of the corresponding order-ααα sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the KKK-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1).
The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of orders∈(1,2]s\in (1,2]s∈(1,2], which has not been addressed previously in the quantum setting.
The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding.