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

• URL: http://arxiv.org/abs/2502.01611v1
• Date: Mon, 03 Feb 2025 18:44:59 GMT
• Title: Additivity and chain rules for quantum entropies via multi-index Schatten norms
• Authors: Omar Fawzi, Jan Kochanowski, Cambyse Rouzé, Thomas Van Himbeeck,
• Abstract summary: We establish a general additivity statement for the optimized sandwiched R'enyi entropy of quantum channels. As an application, we strengthen the additivity statement of [Van Himbeeck and Brown, 2025] thus allowing the analysis of time-adaptive quantum cryptographic protocols.
• Score: 6.574756524825567
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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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