Additivity and chain rules for quantum entropies via multi-index Schatten norms

TL;DR

This paper proves additivity properties of quantum entropies using multi-index Schatten norms, with implications for quantum cryptography and entropy chain rules, advancing understanding of quantum information measures.

Contribution

It generalizes additivity results for sandwiched Rényi entropy to multi-index Schatten norms and establishes chain rules for Rényi conditional entropies.

Findings

Proved additivity of optimized sandwiched Rényi entropy for quantum channels.

Strengthened previous additivity results, enabling analysis of adaptive quantum cryptographic protocols.

Established chain rules for Rényi conditional entropies similar to entropy accumulation theorems.

Abstract

The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched R\'enyi entropy of quantum channels. For that, we generalize the results of [Devetak, Junge, King, Ruskai, CMP 2006] to multi-index Schatten norms. As an application, we strengthen the additivity statement of [Van Himbeeck and Brown, 2025] thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for R\'enyi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].