Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large deviation analysis for quantum security via smoothing of Renyi entropy of order 2

Masahito Hayashi

TL;DR

The paper develops a large-deviation (exponential-rate) analysis of quantum privacy amplification by leveraging smoothing of Rényi entropy of order $2$, providing upper bounds on $L_1$ distinguishability and the modified mutual information under universal$_2$ and $\varepsilon$-almost dual universal$_2$ hash functions. It derives single-shot bounds and asymptotic exponents, proves tightness in a Pauli-channel scenario, and extends the results to secret-key generation with and without error correction, including non-i.i.d. scenarios. By introducing and relating various hash ensembles (dual-universal, permuted-code, $\delta$-biased) and new entropy quantities, the work generalizes Renner’s hashing lemmas to the dual-universal setting and demonstrates improved exponents over min-entropy-based methods in quantum security. The findings have implications for quantum key distribution and privacy amplification protocols, offering theoretically solid, efficiently computable bounds with potential practical benefits in finite-size regimes. Overall, the paper advances quantum information theory by connecting Rényi-entropy smoothing to exponential secrecy guarantees under broad hashing strategies and channel models.

Abstract

It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal$_2$ hash functions. Using the smoothing of Renyi entropy of order 2, we derive security bounds for $L_1$ distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply $\varepsilon$-almost dual universal$_2$ hash functions. Further, we apply this analysis to the secret key generation with error correction.

Large deviation analysis for quantum security via smoothing of Renyi entropy of order 2

TL;DR

The paper develops a large-deviation (exponential-rate) analysis of quantum privacy amplification by leveraging smoothing of Rényi entropy of order , providing upper bounds on distinguishability and the modified mutual information under universal and -almost dual universal hash functions. It derives single-shot bounds and asymptotic exponents, proves tightness in a Pauli-channel scenario, and extends the results to secret-key generation with and without error correction, including non-i.i.d. scenarios. By introducing and relating various hash ensembles (dual-universal, permuted-code, -biased) and new entropy quantities, the work generalizes Renner’s hashing lemmas to the dual-universal setting and demonstrates improved exponents over min-entropy-based methods in quantum security. The findings have implications for quantum key distribution and privacy amplification protocols, offering theoretically solid, efficiently computable bounds with potential practical benefits in finite-size regimes. Overall, the paper advances quantum information theory by connecting Rényi-entropy smoothing to exponential secrecy guarantees under broad hashing strategies and channel models.

Abstract

It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal hash functions. Using the smoothing of Renyi entropy of order 2, we derive security bounds for distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply -almost dual universal hash functions. Further, we apply this analysis to the secret key generation with error correction.

Paper Structure

This paper contains 50 sections, 44 theorems, 244 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Generalization of main results
  5. Relation with second order analysis
  6. Organization
  7. Preparation
  8. Information quantities for single system
  9. Case of sub-states
  10. Case of normalized states
  11. Information quantities in composite system
  12. Case of joint sub-state
  13. Case of joint normalized state
  14. Criteria for secret random numbers
  15. Case of joint sub-state
  16. ...and 35 more sections

Key Result

Lemma 1

The functions $s \mapsto \psi(s|\rho\|\sigma), \underline{\psi}(s|\rho\|\sigma)$ are convex. In particular, they are strictly convex when $\rho$ and $\sigma$ are not completely mixed.

Figures (1)

  • Figure 1: Lower bounds of exponent. Thick dashed line: $e_{\mathop{\rm G},\mathop{\rm q}}(\rho_{A,E}|\log p -R_2) = \max_{0\le s \le 1} \frac{s}{2} (R_2 -H_{1-\frac{s}{1+s}}(X|P_{X}))$ Normal dashed line: $e_{\mathop{\rm H},\mathop{\rm q}}(\rho_{A,E}|\log p -R_2) =\max_{0\le s \le 1} \frac{s}{2-s} (R_2 -H_{1-s}(X|P_{X})) = \max_{0\le t \le 1} \frac{t}{2} (R_2 -H_{1-\frac{2t}{2+t}}(X|P_{X}))$ Thick line: $-\frac{1}{n}\log \min_{0\le s \le 1}$ (RHS of (\ref{['1-2-1']})), Normal line: $-\frac{1}{n}\log \min_{0\le s \le 1}$ (RHS of (\ref{['12-24-1-q']})) with $n=10,000$, $p=2$, $P_X(0)=0.9$, $P_X(1)=0.1$.

Theorems & Definitions (58)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • proof
  • Lemma 9
  • ...and 48 more