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arXiv · 2502.01611

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

Omar Fawzi·Jan Kochanowski·Cambyse Rouzé·Thomas Van Himbeeck
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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].

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Omar Fawzi, Jan Kochanowski, Cambyse Rouzé, Thomas Van Himbeeck. 2025-02-03. Additivity and chain rules for quantum entropies via multi-index Schatten norms. https://doi.org/10.1007/s00220-026-05567-8

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