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

TL;DR

This paper derives the exact exponential decay rate for smoothing the max-relative entropy in quantum information, applies it to quantum privacy amplification, and explores security exponents and equivocation asymptotics.

Contribution

It provides the first exact security exponent for quantum privacy amplification at high rates and analyzes the tightness of bounds across different rate regimes.

Findings

Exact exponent for smoothing max-relative entropy derived.

Upper bound for insecurity decay rate in privacy amplification obtained.

In low-rate cases, bounds are not tight, similar to channel coding error exponents.

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