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

Omar Fawzi, Jan Kochanowski, Cambyse Rouzé, Thomas Van Himbeeck

TL;DR

This work develops a unifying, operator-space-based framework for quantum entropies by employing multi-index Schatten norms, enabling general additivity and chain-rule results for optimized sandwiched Rényi entropies. Central technical advances include a multiplicativity theorem for completely bounded norms across multi-index spaces and corresponding additivity results for output Rényi entropies, together with chain-rule inequalities that extend existing entropy accumulation approaches. By translating these norm-based statements into entropy inequalities, the authors obtain time-adaptive security proofs for quantum cryptographic protocols under time-varying noise, including QRNG and BB84 scenarios. The results have practical impact for cryptographic security under realistic, non-static conditions and broaden the toolbox for analyzing quantum information tasks through a functional-analytic lens that links operator-space theory with quantum entropy. Overall, the paper provides a rigorous path from deep operator-norm results to concrete, time-adaptive cryptographic security guarantees.

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 Rényi 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 Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].

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

TL;DR

This work develops a unifying, operator-space-based framework for quantum entropies by employing multi-index Schatten norms, enabling general additivity and chain-rule results for optimized sandwiched Rényi entropies. Central technical advances include a multiplicativity theorem for completely bounded norms across multi-index spaces and corresponding additivity results for output Rényi entropies, together with chain-rule inequalities that extend existing entropy accumulation approaches. By translating these norm-based statements into entropy inequalities, the authors obtain time-adaptive security proofs for quantum cryptographic protocols under time-varying noise, including QRNG and BB84 scenarios. The results have practical impact for cryptographic security under realistic, non-static conditions and broaden the toolbox for analyzing quantum information tasks through a functional-analytic lens that links operator-space theory with quantum entropy. Overall, the paper provides a rigorous path from deep operator-norm results to concrete, time-adaptive cryptographic security guarantees.

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 Rényi 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 Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].

Paper Structure

This paper contains 35 sections, 26 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Multiplicativity of completely bounded norms
  4. Additivity of output $\alpha$-Rényi conditional entropy
  5. Chain rule for the $\alpha$-Rényi entropy:
  6. Application to quantum cryptography
  7. Structure of the paper
  8. Preliminaries
  9. Basic notation
  10. Operator spaces
  11. Norms on operator-valued Schatten spaces
  12. Norms on linear maps
  13. Rényi conditional entropies
  14. Generalized variational expressions for Pisier norms
  15. Main results: Chain rules and additivity
  16. ...and 20 more sections

Key Result

Proposition 2.1

The family of norms defined above satisfies the following two main properties, namely for any $m,n\in\mathbb{N}$, where $X\oplus Y= $, and $FXG\equiv (F\otimes \mathbbm{1}_\mathcal{K})X(G\otimes \mathbbm{1}_\mathcal{K})$.

Figures (1)

  • Figure 1: The above figure illustrates the main implications presented in this work, excluding the applications to QKD. Violet boxes represent external results, blue boxes our main theorems presented in terms of channel norms, and teal boxes their transcription in terms of conditional Rényi-entropies.

Theorems & Definitions (56)

  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.3: Pisier's formula Book.Pisier.1998
  • Corollary 2.4
  • proof
  • Theorem 2.5: Theorem 4.5 in Book.Pisier.1998
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 46 more