# On the Second-Order Asymptotics of the Partially Smoothed Conditional Min-Entropy & Application to Quantum Compression

## TL;DR

This paper derives the second-order asymptotic expansion of the partially smoothed conditional min-entropy for pure states, revealing state-dependent differences from the globally smoothed case, and applies this to optimize quantum data compression.

## Contribution

It provides the first second-order expansion for partially smoothed conditional min-entropy in the i.i.d. setting and connects this to quantum compression protocols.

## Key findings

- Second-order term differs for pure states compared to globally smoothed entropy.
- The derived expansion determines the optimal quantum data compression rate.
- Straightforward eigenspace cutoff protocol is asymptotically optimal at second order.

## Abstract

Recently, Anshu et al. introduced "partially" smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [arXiv:1807.05630 [quant-ph]]. Yet, a tight second-order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original "globally" smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order term for partially and globally smoothed quantities coincides. By relating the task of quantum compression to that of quantum state merging, our derived expansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.