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Smooth min-entropy lower bounds for approximation chains

Ashutosh Marwah, Frédéric Dupuis

TL;DR

This work develops a framework for lower-bounding the smooth min-entropy of quantum states that form ε-approximation chains, by introducing an entropic triangle inequality that ties $H^{\epsilon}_{\min}$ to Rényi entropies and $D^{\epsilon}_{\max}$. Leveraging this, the authors address approximate independence and approximate entropy accumulation, deriving one-shot-to-asymptotic bridges and providing approximate AEP and EA results, with applications to quantum cryptography. They introduce mixed-channel techniques and a sharp Rényi-divergence formalism to bound divergences between nearby channels and to enable an approximate EA theorem, including testing-based refinements inspired by DIQKD. While offering a powerful conceptual and technical toolkit for non-i.i.d. and correlated-source cryptography, the work also highlights limitations in ε-sensitivity and marks avenues for tighter divergence bounds and improved practical thresholds.

Abstract

For a state $ρ_{A_1^n B}$, we call a sequence of states $(σ_{A_1^k B}^{(k)})_{k=1}^n$ an approximation chain if for every $1 \leq k \leq n$, $ρ_{A_1^k B} \approx_εσ_{A_1^k B}^{(k)}$. In general, it is not possible to lower bound the smooth min-entropy of such a $ρ_{A_1^n B}$, in terms of the entropies of $σ_{A_1^k B}^{(k)}$ without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the Rényi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove approximate versions of the asymptotic equipartition property and entropy accumulation. In our companion paper, we show that the techniques developed in this paper can be used to prove the security of quantum key distribution in the presence of source correlations.

Smooth min-entropy lower bounds for approximation chains

TL;DR

This work develops a framework for lower-bounding the smooth min-entropy of quantum states that form ε-approximation chains, by introducing an entropic triangle inequality that ties to Rényi entropies and . Leveraging this, the authors address approximate independence and approximate entropy accumulation, deriving one-shot-to-asymptotic bridges and providing approximate AEP and EA results, with applications to quantum cryptography. They introduce mixed-channel techniques and a sharp Rényi-divergence formalism to bound divergences between nearby channels and to enable an approximate EA theorem, including testing-based refinements inspired by DIQKD. While offering a powerful conceptual and technical toolkit for non-i.i.d. and correlated-source cryptography, the work also highlights limitations in ε-sensitivity and marks avenues for tighter divergence bounds and improved practical thresholds.

Abstract

For a state , we call a sequence of states an approximation chain if for every , . In general, it is not possible to lower bound the smooth min-entropy of such a , in terms of the entropies of without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the Rényi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove approximate versions of the asymptotic equipartition property and entropy accumulation. In our companion paper, we show that the techniques developed in this paper can be used to prove the security of quantum key distribution in the presence of source correlations.
Paper Structure (19 sections, 29 theorems, 236 equations, 3 figures)

This paper contains 19 sections, 29 theorems, 236 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Entropic triangle inequality for the smooth min-entropy
  4. Approximately independent registers
  5. Weak approximate asymptotic equipartition
  6. Simple security proof for sequential device independent quantum key distribution
  7. Approximate entropy accumulation
  8. Divergence bound for approximately equal states
  9. Bounding the channel divergence for two channels close to each other
  10. Proof of the approximate entropy accumulation theorem
  11. Testing
  12. Limitations and further improvements
  13. Entropic triangle inequalities cannot be improved much
  14. Bounds for $D^{\#}_{\alpha}$ of the form in Lemma \ref{['lemm:Dsharp_close_st_bd']} necessarily diverge in the limit $\alpha=1$
  15. Transforming lemmas for EAT from $\tilde{H}^{\downarrow}_\alpha$ to $\tilde{H}^{\uparrow}_\alpha$
  16. ...and 4 more sections

Key Result

Lemma 3.1

Let $\rho$ and $\eta$ be subnormalised states and $Q$ be a positive operator, then for $\alpha>1$, we have and for $\alpha<1$ if one of $\tilde{D}_{\alpha}(\eta || Q)$ and $D_{\max}(\rho || \eta)$ is finite (otherwise we cannot define their difference), we have

Figures (3)

  • Figure 1: The lower bound in Eq. \ref{['eq:single_rnd_bnd']} for the interval $[\frac{3}{4}, \frac{2+\sqrt{2}}{4}]$
  • Figure 2: The setting for entropy accumulation and Theorem \ref{['th:approx_EAT']}. For $k \in [n]$, the channels $\mathop{\mathrm{\mathcal{M}}}\nolimits_k$ are repeatedly applied to the registers $R_{k-1}$ to produce the "secret" information $A_k$ and the side information $B_k$.
  • Figure 3: Setting for classical EAT

Theorems & Definitions (57)

  • Lemma 3.1
  • proof
  • Definition 3.2: $\epsilon$-smooth $\alpha$-Rényi conditional entropy
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • ...and 47 more