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Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

Bartosz Regula, Marco Tomamichel

Abstract

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. For the key case of measured smooth Rényi divergence of order 2, we show that this can be alternatively understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on quantum privacy amplification and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of randomness extraction, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, which in the asymptotic i.i.d. limit recover the state-of-the-art error exponent of [Dupuis, arXiv:2105.05342]. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.

Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

Abstract

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. For the key case of measured smooth Rényi divergence of order 2, we show that this can be alternatively understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on quantum privacy amplification and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of randomness extraction, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, which in the asymptotic i.i.d. limit recover the state-of-the-art error exponent of [Dupuis, arXiv:2105.05342]. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
Paper Structure (23 sections, 26 theorems, 95 equations, 1 figure)

This paper contains 23 sections, 26 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Preliminaries
  4. Relative entropies and conditional entropies
  5. Randomness extraction and privacy amplification
  6. Measured smooth collision divergence
  7. Motivation: the subtle question of smoothing
  8. Measured collision divergence
  9. Variational form through Hermitian smoothing
  10. One-shot divergence inequalities
  11. Achievability for privacy amplification
  12. Leftover hashing with measured smooth collision divergence
  13. From smoothing to Rényi divergences and error exponents
  14. Fully quantum extension to decoupling
  15. Converse bounds for privacy amplification
  16. ...and 8 more sections

Key Result

Theorem 2

For all quantum states $\rho$, all operators $\sigma \geq 0$, and all $\varepsilon \in [0,1)$, it holds that where $(x)_+ = \max \{ 0, x \}$, and for consistency we understand $a / 0 = \infty \; \forall a > 0$ and $0 / 0 = 0$. Furthermore, $D_{2} ^{\varepsilon,\mspace{2mu} \IfEqCase{m}{ {t}{ T } {p}{ P } {m}{ {\mathbb{M}} } } } (\rho\|\sigma) < \infty$ if and only if $\mathop{\ma

Figures (1)

  • Figure 1: When defining quantum divergences, the operations of smoothing (optimising a divergence over distributions in an $\varepsilon$-ball of trace distance) and measuring (taking the minimal quantum extension of a classical divergence by maximising it over all measurement channels) do not commute. In this work we show that it is the quantity $D_{\max} ^{\varepsilon,\mspace{2mu} \IfEqCase{m}{ {t}{ T } {p}{ P } {m}{ {\mathbb{M}} } } }$, obtained by first smoothing the classical divergence and only then lifting it to quantum states, that more tightly characterises randomness extraction.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 2
  • Remark
  • Lemma 3: Diverging case
  • Lemma 4: Strong duality
  • Lemma 5: Classical case
  • Lemma 6: Data processing
  • Corollary 7
  • Lemma 8
  • Lemma 9
  • ...and 20 more