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

Abstract

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 commentV3: close to published version

References in corpus (41)

• Distillation of secret key and entanglement from quantum states
• On quantum Renyi entropies: a new generalization and some properties
• Distance measures to compare real and ideal quantum processes
• Noncommuting mixed states cannot be broadcast
• The Quantum Chernoff Bound
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• Leftover Hashing Against Quantum Side Information
• One-Shot Classical-Quantum Capacity and Hypothesis Testing
• A Fully Quantum Asymptotic Equipartition Property
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Duality Between Smooth Min- and Max-Entropies
• Exponential decreasing rate of leaked information in universal random privacy amplification
• Monotonicity of a relative Rényi entropy
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
• Second-order asymptotics for quantum hypothesis testing
• An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
• Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
• Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing
• Finite blocklength converse bounds for quantum channels
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• A reversible theory of entanglement and its relation to the second law
• Strong converse exponent for classical-quantum channel coding
• Smooth Renyi Entropies and the Quantum Information Spectrum
• Relating different quantum generalizations of the conditional Renyi entropy
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Tight exponential analysis of universally composable privacy amplification and its applications
• Generalized Entropies
• Security analysis of epsilon-almost dual universal2 hash functions: smoothing of min entropy vs. smoothing of Rényi entropy of order 2
• Large deviation analysis for quantum security via smoothing of Renyi entropy of order 2
• Strong converse theorems using Rényi entropies
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Precise evaluation of leaked information with universal2 privacy amplification in the presence of quantum attacker
• Partially smoothed information measures
• Two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states

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• Quantum Rényi and $f$-divergences from integral representations
• Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels
• Reliability Function of Classical-Quantum Channels
• Geometric relative entropies and barycentric Rényi divergences
• Tight lower bound on the error exponent of classical-quantum channels
• Reliable Simulation of Quantum Channels: the Error Exponent
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels
• On distinguishability distillation and dilution exponents
• Tight relations and equivalences between smooth relative entropies