Measured Smooth Entropies

• Measured smooth entropies are operational measures that refine classical and quantum Rényi entropies through smoothing of nearby states and measurement lifting.
• They underpin finite blocklength tasks—such as privacy amplification, channel simulation, and source coding—by enabling tight one-shot and second-order analyses.
• Their key properties, including data-processing monotonicity and hypothesis-testing duality, improve the precision and security of both quantum and classical information protocols.

Measured smooth entropies are a class of operationally motivated entropy measures originating from the quantum and classical one-shot information theory literature. They refine standard (Rényi) entropic quantities via two key mechanisms: (1) optimization/smoothing over nearby sub-normalized states or probability distributions, and (2) lifting of classical divergence and entropy formulations to the quantum setting via measurement optimization. These entropies, including measured smooth max-relative entropy and measured smooth conditional Rényi entropy, play a central role in quantifying resources in tasks such as privacy amplification, data compression, and channel simulation, especially under non-asymptotic (finite blocklength) constraints (Tomamichel et al., 2010, Regula et al., 21 Jan 2025, Regula et al., 4 Mar 2026, Sakai et al., 2020).

1. Formal Definitions and Construction

Measured smooth entropies generalize classical and quantum Rényi entropies by introducing smoothing (allowing small perturbations in total variation/trace/purified distance) and measurement lifting.

• Measured Smooth Max-Relative Entropy:

Given quantum states $\rho, \sigma$ on a finite-dimensional Hilbert space, and $\varepsilon \in [0,1]$, the measured smooth max-relative entropy is

$$D_{\max,\,\mathbb{M}}^\varepsilon(\rho\|\sigma) = \sup_{M \in \mathbb{M}} D_{\max}^\varepsilon\bigl(p_{\rho, M} \| p_{\sigma, M}\bigr) = \widetilde D_{\max}^\varepsilon(\rho\|\sigma)$$

where $D_{\max}^\varepsilon(p\|q)$ is the smoothed max-relative entropy between classical distributions $p, q$ and $p_{\rho, M}$ is the POVM-induced distribution. The $\widetilde D_{\max}^\varepsilon$ "information-spectrum" form is

$\widetilde D_{\max}^\varepsilon(\rho\|\sigma) = \log \min\{\lambda \geq 0 : \Tr[\rho - \lambda\sigma]_+ \leq \varepsilon\}$

(Regula et al., 21 Jan 2025).

• Measured Smooth Conditional Rényi Entropy:

For a classical-quantum state $\rho_{XE}$ and order $\alpha > 1$,

$H^{\varepsilon, \mathbb{M}, \uparrow}_\alpha(X|E)_\rho = -\inf_{\sigma_E > 0, \Tr \sigma_E = 1} D^{\varepsilon, \mathbb{M}}_\alpha(\rho_{XE} \| I_X \otimes \sigma_E)$

with the measured smooth divergence

$$D^{\varepsilon, \mathbb{M}}_\alpha(\rho\|\sigma) = \sup_{M\in\mathbb{M}} D^{\varepsilon,T}_\alpha\bigl(M(\rho) \| M(\sigma)\bigr)$$

(Regula et al., 4 Mar 2026).

• Smoothing Paradigm:

Smoothing is performed by optimizing over nearby states $\rho'$ with $\|\rho' - \rho\| \leq \varepsilon$ (trace, purified, or total variation distance) or, for measured or information-spectrum variants, via truncation or sub-operators constrained in distance.

2. Key Properties and Theoretical Relations

Measured smooth entropies satisfy several information-theoretic properties central for operational tasks:

1. Data-Processing Monotonicity: For any CPTP map $\Phi$,

$$D^{\varepsilon, \mathbb{M}}_\alpha(\Phi(\rho)\|\Phi(\sigma)) \leq D^{\varepsilon, \mathbb{M}}_\alpha(\rho\|\sigma)$$

(Regula et al., 4 Mar 2026).

2. Duality and Information Spectrum Equivalence: Measured smooth max-relative entropies are in exact two-way equivalence with hypothesis-testing relative entropy, facilitating tight conversion between achievability and converse in one-shot settings:

$$D_H^{1-\varepsilon}(\rho\|\sigma) = \inf_{0 \leq \delta < \varepsilon} \{\widetilde D_{\max}^\delta(\rho\|\sigma) - \log(\varepsilon-\delta) \}$$

$$\widetilde D_{\max}^\varepsilon(\rho\|\sigma) = \sup_{\varepsilon < \delta \leq 1} \{D_H^{1-\delta}(\rho\|\sigma) + \log(\delta-\varepsilon)\}$$

(Regula et al., 21 Jan 2025).

3. Interpolation and Large Deviations: For $\alpha \in (1,2]$,

$$D^{\varepsilon, \mathbb{M}}_2 \leq (\alpha-1)D^{\mathbb{M}}_\alpha + (2-\alpha)D^{\varepsilon, \mathbb{M}}_{\max}$$

supporting exponential error bounds in hypothesis testing and privacy amplification (Regula et al., 4 Mar 2026).

4. Tightened Gentle Measurement and Datta–Renner Lemmas:

Improvements to fundamental lemmas underpin tighter fidelity and trace-distance approximations: - For $0 \leq M \leq I,\, \Tr[M\rho] \geq 1-\varepsilon$, then the fidelity

$F(\rho, \sqrt{M}\rho\sqrt{M}) \geq (1-\varepsilon)^2, \quad \tfrac12 \|\rho - \tfrac{\sqrt{M}\rho\sqrt{M}}{\Tr M\rho}\|_1 \leq \sqrt{\varepsilon}$

(Regula et al., 21 Jan 2025).

3. Measured Smooth Rényi Entropy in Classical and Quantum Regimes

Measured smooth Rényi entropy generalizes both the operational and asymptotic behavior of smooth entropies across classical and quantum settings:

• Classical (Commuting) Case:

For commuting states, measured smoothing reduces to classical information-spectrum quantities, and trace- or total-variation smoothing suffices (Regula et al., 21 Jan 2025, Sakai et al., 2020). The asymptotics yield direct strong-converse exponent relations and second-order refinements.

• Quantum Measurement Lifting:

The quantum version is defined by the supremum over all POVMs applied to $\rho, \sigma$, capturing the optimal information extractable via measurement. This is crucial for operationalizing entropy in scenarios where quantum side-information persists, such as privacy amplification and decoupling (Regula et al., 4 Mar 2026).

• Conditional and One-Shot Forms:

Conditional measured smooth entropies appear by minimizing over states on side-information systems, yielding tight one-shot characterizations for tasks involving eavesdroppers and adversaries.

4. Operational Applications: Privacy Amplification, Coding, and Guessing

Measured smooth entropies are central to one-shot and finite-blocklength protocols in information theory and quantum cryptography:

• Privacy Amplification:

The measured-smooth leftover hash lemma demonstrates that the extractable key length up to trace distance $\varepsilon$ is

$$\ell_\varepsilon(\rho_{XE}) \geq H^{2,\mathbb{M},\uparrow}_{\varepsilon-\mu}(X|E) - \ln\frac{1}{4\mu^2}$$

surpassing previous smooth-min-entropy-based results, and achieving optimality up to logarithmic terms (Regula et al., 4 Mar 2026).

• Channel Simulation and Resource Transformation:

The achievability/converse duality implies that every one-shot coding bound in terms of observed smooth entropies can be tightly matched to hypothesis-testing converses and vice versa (Regula et al., 21 Jan 2025).

• Source Coding and Guessing:

Asymptotic expansions for smooth Rényi entropies translate directly to precise second-order and moderate deviation bounds for variable-length source coding, task encoding, and guessing with giving-up (Sakai et al., 2020).

Task	Key Measured Smooth Entropy	Reference
Privacy Amplification	$H^{2,\mathbb{M},\uparrow}_{\varepsilon}$	(Regula et al., 4 Mar 2026)
Channel Simulation	$D^{\varepsilon, \mathbb{M}}_{\max}$, $D_H$	(Regula et al., 21 Jan 2025)
Source Coding	$H^{\varepsilon}_{\alpha}(X|Y)$	(Sakai et al., 2020)
Guessing	$H^{\varepsilon}_{\alpha}(X|Y)$, $ċH^{\varepsilon}_{\alpha}$	(Sakai et al., 2020)

Tight duality and moderate deviation analysis facilitate the precise quantification of security, efficiency, and reliability under resource constraints.

5. Asymptotic Expansions and Second-Order Analysis

Measured smooth entropies exhibit refined asymptotics for i.i.d. sources:

• Second-Order Expansions:

For $\rho^{\otimes n}$, the measured smooth Rényi (especially for $\alpha=2$) matches the hypothesis-testing divergence up to $O(\ln n)$, yielding for privacy amplification:

$$\ell_\varepsilon(\rho_{XE}^{\otimes n}) = n H(X|E) + \sqrt{n V(X|E)} \Phi^{-1}(\varepsilon) + O(\ln n)$$

(Regula et al., 4 Mar 2026).

• Large-Deviation Regime:

The error exponent for key extraction or error probabilities is determined by the measured Rényi spectrum:

$$\liminf_{n\to\infty} -\frac{1}{n}\ln \varepsilon_n \geq \sup_{\alpha \in (1,2]}\frac{\alpha-1}{\alpha} \bigl(\widetilde H_\alpha^{\uparrow}(X|E)-R\bigr)$$

(Regula et al., 4 Mar 2026).

6. Broader Structural Properties and Dynamical Generalizations

Sub-additivity, concavity, and additivity properties are inherited by measured smooth entropies under composition and product structure. The extension of entropy sub-additivity to commuting smooth transformations on Banach spaces (Luo et al., 2022) demonstrates that the essential monotonicity and structural features of entropies are preserved even in infinite-dimensional dynamical contexts, underpinning their robustness as measures of disorder and resource.

7. Significance and Impact

Measured smooth entropies unify operational one-shot information measures with asymptotic, spectrum-based approaches, eliminating previous technical gaps between achievability and converse. They yield:

• The tightest known one-shot and second-order characterizations under trace distance for quantum cryptographic and information-theoretic tasks (Regula et al., 4 Mar 2026).
• A precise map between operational tasks governed by different entropy types (e.g., covering vs. packing protocols) (Regula et al., 21 Jan 2025).
• Enhanced protocols for privacy amplification and decoupling, optimal up to constant-order corrections even in quantum settings.
• Theoretical tools for the analysis of entropy in smooth dynamical systems and infinite-dimensional analysis (Luo et al., 2022).

Measured smooth entropies thus form a foundational toolkit for modern quantum and classical information theory, bridging operational security, coding, and resource conversion in both finite and asymptotic domains.