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.

V3: close to published version

References in corpus (7)

• Distillation of secret key and entanglement from quantum states
• The Quantum Chernoff Bound
• Leftover Hashing Against Quantum Side Information
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
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Cited by in corpus (10)

• 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