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Operational Interpretation of the Sandwiched Rényi Divergence of Order 1/2 to 1 as Strong Converse Exponents

Ke Li, Yongsheng Yao

TL;DR

This work provides the first operational interpretation of the sandwiched Rényi divergence $D_{\alpha}^{*}$ for $\alpha\in(\tfrac{1}{2},1)$ by deriving exact strong converse exponents for smoothing of the max-relative entropy, quantum privacy amplification, and quantum information decoupling. The exponents are expressed through $D_{\alpha}^{*}$, its induced conditional entropies $H_{\alpha}^{*}$, and the regularized sandwiched Rényi mutual information $I_{\alpha}^{*,\mathrm{reg}}$, reinforcing the duality between the intervals $\alpha\in(\tfrac{1}{2},1)$ and $\alpha\in(1,\infty)$. The paper uses two complementary methods: (i) a type-based (commutative) analysis followed by a fidelity-based lifting to the general quantum case for smoothing, and (ii) a variational-programmatic approach to achievability and a tight converse for privacy amplification and decoupling, relying on both log-Euclidean and sandwiched Rényi quantities. The results extend the operational significance of $D_{\alpha}^{*}$ to the $\alpha\in(\tfrac{1}{2},1)$ regime and highlight the nuanced role of regularized information measures in quantum information tasks. The findings have implications for reliability functions and non-asymptotic quantum information processing, and they raise open questions about trace-distance exponents and single-letter characterizations in decoupling.

Abstract

We provide the sandwiched Rényi divergence of order $α\in(\frac{1}{2},1)$, as well as its induced quantum information quantities, with an operational interpretation in the characterization of the exact strong converse exponents of quantum tasks. Specifically, we consider (a) smoothing of the max-relative entropy, (b) quantum privacy amplification, and (c) quantum information decoupling. We solve the problem of determining the exact strong converse exponents for these three tasks, with the performance being measured by the fidelity or purified distance. The results are given in terms of the sandwiched Rényi divergence of order $α\in(\frac{1}{2},1)$, and its induced quantum Rényi conditional entropy and quantum Rényi mutual information. This is the first time to find the precise operational meaning for the sandwiched Rényi divergence with Rényi parameter in the interval $α\in(\frac{1}{2},1)$.

Operational Interpretation of the Sandwiched Rényi Divergence of Order 1/2 to 1 as Strong Converse Exponents

TL;DR

This work provides the first operational interpretation of the sandwiched Rényi divergence for by deriving exact strong converse exponents for smoothing of the max-relative entropy, quantum privacy amplification, and quantum information decoupling. The exponents are expressed through , its induced conditional entropies , and the regularized sandwiched Rényi mutual information , reinforcing the duality between the intervals and . The paper uses two complementary methods: (i) a type-based (commutative) analysis followed by a fidelity-based lifting to the general quantum case for smoothing, and (ii) a variational-programmatic approach to achievability and a tight converse for privacy amplification and decoupling, relying on both log-Euclidean and sandwiched Rényi quantities. The results extend the operational significance of to the regime and highlight the nuanced role of regularized information measures in quantum information tasks. The findings have implications for reliability functions and non-asymptotic quantum information processing, and they raise open questions about trace-distance exponents and single-letter characterizations in decoupling.

Abstract

We provide the sandwiched Rényi divergence of order , as well as its induced quantum information quantities, with an operational interpretation in the characterization of the exact strong converse exponents of quantum tasks. Specifically, we consider (a) smoothing of the max-relative entropy, (b) quantum privacy amplification, and (c) quantum information decoupling. We solve the problem of determining the exact strong converse exponents for these three tasks, with the performance being measured by the fidelity or purified distance. The results are given in terms of the sandwiched Rényi divergence of order , and its induced quantum Rényi conditional entropy and quantum Rényi mutual information. This is the first time to find the precise operational meaning for the sandwiched Rényi divergence with Rényi parameter in the interval .
Paper Structure (25 sections, 26 theorems, 208 equations)

This paper contains 25 sections, 26 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Quantum Rényi divergences and entropies
  5. Method of types
  6. Problems and Main Results
  7. Smoothing of the max-relative entropy
  8. Quantum privacy amplification
  9. Quantum information decoupling
  10. Proof of the Strong Converse Exponent for Smoothing the Max-Relative Entropy
  11. Variational expression
  12. Commutative case
  13. General case
  14. Proof of the Strong Converse Exponent for Quantum Privacy Amplification
  15. Variational expression
  16. ...and 10 more sections

Key Result

Lemma 1

For a quantum system $A$ of finite dimension and any $n\in\mathbb{N}$, there exists a single symmetric state $\sigma_{A^n}^u \in \mathcal{S}_{\rm{sym}}(A^n)$ such that every symmetric state $\sigma_{A^n} \in \mathcal{S}_{\rm{sym}}(A^n)$ is dominated as where $v_{n,|A|} \leq (n+1)^{\frac{(|A|+2)(|A|-1)}{2}}$ is a polynomial of $n$. The number of distinct eigenvalues of $\sigma_{A^n}^u$ is upper bo

Theorems & Definitions (36)

  • Lemma 1
  • Definition 2
  • Remark 3
  • Proposition 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Theorem 8
  • Proposition 9
  • Remark 10
  • ...and 26 more