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.

• The paper introduces a new class of measured smooth entropies that sharpen one-shot bounds in quantum privacy amplification.
• It establishes improved randomness extraction rates and error exponents through a novel smoothing paradigm over Hermitian operators.
• The work rigorously derives tight second-order asymptotics and converse bounds, refining the operational understanding of quantum security.

Tight One-Shot Analysis of Quantum Privacy Amplification via Measured Smooth Entropies

Motivation and Background

Quantum privacy amplification is central in establishing the security of quantum key distribution (QKD) and more broadly in applications where secret randomness must be distilled in the presence of adversarial quantum side information. Classical privacy amplification has traditionally been characterized by the leftover hash lemma, which relates the extractable randomness to min-entropy and collision entropy of the randomness source. In quantum information theory, these concepts are generalized via quantum Rényi entropies and divergences, notably the sandwiched and Petz--Rényi divergences, as well as their smooth variants. Smoothing, i.e., optimizing over states close (in trace or purified distance) to the true source, has enabled sharper characterizations, especially in the classical regime.

Quantum settings present distinct challenges: smoothing techniques often prove suboptimal for quantum adversaries, particularly when operational security criteria rely on trace distance. Previous asymptotic error estimates and refined second-order analyses were only available under modified distance criteria (e.g., purified distance), motivating the search for more refined approaches that recover both sharp second-order and large deviation scaling for all operational meaningful choices of trace distance.

Measured Smooth Entropies and Novel Smoothing Paradigms

This work introduces a new class of measured smooth conditional entropies, founded on lifting classical smooth divergence definitions to quantum states through measurements. The measured smooth Rényi divergence of order 2, $D_2^{\mathbb{M},\varepsilon}$, is highlighted as a richer alternative to the sandwiched Rényi divergence employed in prior works for one-shot bounds. A key realization is that smoothing can be generalized not only over states but also over Hermitian operators (including non-positive ones), echoing classical smoothing over sub-distributions but reflecting the non-lattice structure of positive operators in quantum theory.

The measured smooth collision divergence, defined as

$$D_2^{\mathbb{M},\varepsilon}(\rho\|\sigma) = \sup_{\mathcal{M} \in \mathbb{M}} D_2^{\varepsilon,T}\big(\mathcal{M}(\rho) \| \mathcal{M}(\sigma) \big),$$

is shown to admit a variational characterization allowing smoothing over Hermitian $R \leq \rho$, optimizing the Rényi divergence with additional trace distance constraints. This leads to optimality conditions and a direct connection to classical guessing probability formulations, extending to quantum adversaries and establishing the operational relevance of these new smooth entropies.

Tightened Leftover Hash Lemma and Achievability

Leveraging measured smooth entropies, the authors derive a strictly improved leftover hash lemma for quantum privacy amplification, yielding bounds that outperform all previous smooth min-entropy-based results. Specifically,

$$H_{2}^{\mathbb{M},\varepsilon-\mu}(X|E)_\rho - \log\frac{1}{4\mu^2} \leq \ell_\varepsilon(\rho_{XE}) \leq H_{\min}^{\mathbb{M},\varepsilon+\delta}(X|E)_\rho + \log\frac{\varepsilon+\delta}{\delta},$$

where $\ell_\varepsilon$ is the largest number of random bits extractable to error $\varepsilon$, and the entropy quantities are defined via measured smoothing. Notably, the scaling with $\sqrt{\varepsilon}$ for purified distance smoothing is shown to be tight, and measured smooth min-entropy yields greater extractable randomness, especially in finite-copy regimes.

The result is achieved through an improved smoothing mechanism, allowing optimization over Hermitian operators and connecting these to measured statistics via the Bures inner product, which in turn links to quantum Fisher information and monotone operator metrics.

Asymptotic Rates, Error Exponents, and Decoupling

The measured smooth collision entropy is shown to interpolate smoothly to measured Rényi divergences, thereby recovering the sharpest known error exponents in the i.i.d. regime. For protocol rates $R < H(X|E)_\rho$, the achievable error exponent matches the classical bound, now established for quantum adversaries: $$\liminf_{n\to\infty} -\frac{1}{n}\log(\varepsilon_n) \geq \sup_{\alpha \in (1,2]} \frac{\alpha-1}{\alpha} \left( H_\alpha^{\mathbb{M}}(X|E)_\rho - R \right).$$ This result is obtained without recourse to norm interpolation machinery and improves upon prior bounds based on sandwiched Rényi divergences.

The approach readily generalizes to the decoupling setting (the fully quantum variant of privacy amplification), yielding improved one-shot bounds for randomness extraction from quantum states, tight concentration inequalities, and optimal second-order asymptotics under trace distance.

One-Shot and Asymptotic Converse Bounds

The paper establishes approximate tightness of the achievability bounds by proving matching converse results up to logarithmic terms. Notably, the optimal second-order asymptotic expansion under trace distance is rigorously recovered: $$\ell_\varepsilon(\rho_{XE}^{\otimes n}) = n H(X|E)_\rho + \sqrt{n V(X|E)_\rho} \,\Phi^{-1}(\varepsilon) + O(\log n),$$ where $V(X|E)_\rho$ is the quantum relative entropy variance and $\Phi^{-1}$ is the quantile function of the standard normal distribution.

Furthermore, the work clarifies the limitations of classical-like converse bounds in quantum regimes and settles long-standing gaps in ensemble vs. universal hashing optimality for strong converse exponents and error rates.

Implications and Broader Impact

The findings point to measured smooth min-entropy as the most appropriate quantum generalization of classical smooth min-entropy for characterizing extractable randomness in quantum privacy amplification. The results underscore the need to reassess smoothing paradigms in quantum information theory, as traditional approaches may obscure operational optimality in crucial cryptographic and communication primitives. The implications reach beyond privacy amplification and decoupling, hinting at more efficient and accurate finite-blocklength analyses in quantum coding, cryptography, and hypothesis testing.

Future directions include extending these techniques to operational settings with restricted measurements, computationally bounded distinguishers, and other tasks where measurement-induced smoothing is preferable due to the operational structure of security criteria. Additionally, further study into monotonicity properties and the interplay of global versus local measurement constructions may yield even finer refinements within quantum entropy theory.

Conclusion

This work offers a unified and strictly sharper one-shot and asymptotic analysis of quantum privacy amplification, demonstrating that measured smooth entropies faithfully and optimally characterize extractable randomness in the presence of quantum adversaries. By establishing a new smoothing paradigm—lifting classical smoothing by measurements and permitting Hermitian operator smoothing—the paper not only recovers but strengthens the sharpest known bounds, delivers improved error exponents, and proves optimal second-order rates. The methodological innovations are of practical and theoretical importance, and are likely to catalyze further advancements in operational quantum information theory and cryptography.