Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.10263 · IEEE International Symposium on Information Theory (ISIT)

Strong Converse for Privacy Amplification against Quantum Side Information

Yuchen Shen, Li Gao, Hao–Chung Cheng

Published Jun 26, 2022• 6 citations
arXivHTMLPDF

Abstract

We establish a one-shot strong converse bound for privacy amplification against quantum side information with trace distance as a security criterion. Firstly, our result shows that the trace distance converges to 1 exponentially fast for any finite blocklength when the rate of the extractable randomness exceeds the conditional von Neumann entropy of Alice given Eavesdropper. Secondly, we obtain a second-order converse bound for the maximal extractable randomness. Thirdly, our result extends to the moderate deviation regime. Namely, when the rate of the extractable randomness approaches the conditional von Neumann entropy from above with speed slower than $O\left( {1/\sqrt n } \right.$, the trace distance still converges to 1 asymptotically. Lastly, our result yields strong converse to entropy accumulation, which complements the recent result by Dupuis [arXiv:2105.05342]. The full manuscript can be assessed in [1], [2].

DOI
10.1109/isit50566.2022.9834467

Related papers

Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

Yuchen Shen, Li Gao, Hao–Chung Cheng

We study quantum soft covering and privacy amplification against quantum side information. The former task aims to approximate a quantum state by sampling from a prior distribution and querying a quantum channel. The latter task aims to extract uniform and independent randomness against quantum adversaries. For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state. We show that the minimal amount of samples for achieving an ε-covering is given by the (1 − ε)-hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an ε-secret extractor is characterized by the conditional (1 − ε)-hypothesis testing entropy. When performing independent and identical repetitions of the tasks, our one-shot characterizations lead to tight asymptotic expansions of the above-mentioned operational quantities. We establish their second-order rates given by the quantum mutual information variance and the quantum conditional information variance, respectively. Moreover, our results extend to the moderate deviation regime, which are the optimal asymptotic rates when the trace distances vanish at sub-exponential speed. Our proof technique is direct analysis of trace distance without smoothing.

arXiv:2202.11590•IEEE Transactions on Information Theory•Jan 10, 2024•18 citations

Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

Bartosz Regula, Marco Tomamichel

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.

quant-pharXiv:2603.04493•Mar 6, 2026•0 citations

Total insecurity of communication via strong converse for quantum privacy amplification

Salzmann, Robert, Nilanjana Datta

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.

arXiv:2202.11090•Feb 22, 2022•1 citation

The strong converse exponent of composable randomness extraction against quantum side information

Roberto Rubboli, Marco Tomamichel

We find a tight characterization of the strong converse exponent for randomness extraction against quantum side information. In contrast to previous tight bounds, we employ a composable error criterion given by the fidelity (or purified distance) to a uniform distribution in product with the marginal state. Our characterization is formulated in terms of the club-sandwiched conditional entropy, recently introduced by Rubboli, Goodarzi, and Tomamichel. It extends the corresponding result for classical side information, whose strong converse exponent was first established by Berta and Yao and later recovered by Li, Li, and Yu within this entropic framework. This provides the first precise operational interpretation of this family of conditional entropies in the quantum setting.

quant-pharXiv:2601.19182•IEEE Transactions on Information Theory•Jan 28, 2026•0 citations

Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification

Ke Li, Yongsheng Yao, Masahito Hayashi

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 order$s\in (1,2]$, which has not been addressed previously in the quantum setting.

arXiv:2111.01075•IEEE Transactions on Information Theory•Nov 4, 2022•24 citations

Strong Converse Exponents of Quantum Soft Covering and Privacy Amplification

Shi-Bing Li, Hongsen Qiu, Xinyu Zhang

We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders $α\in[\frac{1}{2},\infty)$. For $α\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $α\in[1,\infty)$, it is characterized by the order-$α$ 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 $α\in(2,\infty)$, expressed in terms of the corresponding order-$α$ 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 $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $α\in[\frac{1}{2},1)$.

quant-pharXiv:2608.16452•Aug 18, 2026•0 citations
We establish a one-shot strong converse bound for privacy amplification against quantum side information with trace distance as a security criterion. Firstly, our result shows that the trace distance converges to 1 exponentially fast for any finite blocklength when the rate of the extractable randomness exceeds the conditional von Neumann entropy of Alice given Eavesdropper. Secondly, we obtain a second-order converse bound for the maximal extractable randomness. Thirdly, our result extends to the moderate deviation regime. Namely, when the rate of the extractable randomness approaches the conditional von Neumann entropy from above with speed slower than O(1/nO\left( {1/\sqrt n } \right.O(1/n​, the trace distance still converges to 1 asymptotically. Lastly, our result yields strong converse to entropy accumulation, which complements the recent result by Dupuis [arXiv:2105.05342]. The full manuscript can be assessed in [1], [2].
We study quantum soft covering and privacy amplification against quantum side information. The former task aims to approximate a quantum state by sampling from a prior distribution and querying a quantum channel. The latter task aims to extract uniform and independent randomness against quantum adversaries. For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state. We show that the minimal amount of samples for achieving an ε-covering is given by the (1 − ε)-hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an ε-secret extractor is characterized by the conditional (1 − ε)-hypothesis testing entropy. When performing independent and identical repetitions of the tasks, our one-shot characterizations lead to tight asymptotic expansions of the above-mentioned operational quantities. We establish their second-order rates given by the quantum mutual information variance and the quantum conditional information variance, respectively. Moreover, our results extend to the moderate deviation regime, which are the optimal asymptotic rates when the trace distances vanish at sub-exponential speed. Our proof technique is direct analysis of trace distance without smoothing.
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 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 find a tight characterization of the strong converse exponent for randomness extraction against quantum side information. In contrast to previous tight bounds, we employ a composable error criterion given by the fidelity (or purified distance) to a uniform distribution in product with the marginal state. Our characterization is formulated in terms of the club-sandwiched conditional entropy, recently introduced by Rubboli, Goodarzi, and Tomamichel. It extends the corresponding result for classical side information, whose strong converse exponent was first established by Berta and Yao and later recovered by Li, Li, and Yu within this entropic framework. This provides the first precise operational interpretation of this family of conditional entropies 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 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.
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).