Finite-Size Min-Entropy (FME) Approach

• FME is a finite-size measure based on smooth min-entropy that quantifies the adversary's guessing probability in classical-quantum systems.
• It delivers explicit key-rate bounds in QKD by integrating finite statistical effects, error correction leakage, and composable security criteria.
• The method employs parameter estimation, privacy amplification, and SDP optimization to facilitate efficient randomness extraction and secure communication.

Finite-size Min-Entropy (FME) is a central construct in quantum information theory and cryptography, quantifying the unpredictability of classical-quantum systems in the finite-block regime. Unlike asymptotic approaches, FME yields quantitatively precise bounds in realistic scenarios where only a limited number of signals, rounds, or data samples are available. It directly determines operationally meaningful rates for secrecy, randomness extraction, and quantum communication under composable security definitions.

1. Definitions and Operational Meaning

FME is rooted in the (smooth) conditional min-entropy $H_{\min}^\epsilon(X|E)_\rho$ of a classical-quantum state,

$$\rho_{XE} = \sum_{x\in\mathcal{X}} p_X(x) |x\rangle\langle x| \otimes \rho_E^x,$$

where $X$ is a classical register and $E$ encodes quantum side-information. The min-entropy characterizes the adversary's optimal guessing probability,

$$H_{\min}(X|E)_\rho := - \log_2 P_{\text{guess}}(X|E)_\rho, \qquad P_{\text{guess}}(X|E)_\rho = \max_{\{M_x\}} \sum_x p_X(x)\, \mathrm{Tr}\left[M_x\, \rho_E^x\right].$$

The smoothed variant,

$$H_{\min}^\epsilon(X|E)_\rho := \max_{\tilde\rho_{XE}\in B^\epsilon(\rho_{XE})} H_{\min}(X|E)_{\tilde\rho},$$

where $B^\epsilon(\rho_{XE})$ is an $\epsilon$-ball in purified distance, allows dismissal of unlikely events with total probability up to $\epsilon$, granting composable security guarantees. In QKD, FME quantifies the extractable secret key length after error correction and privacy amplification, explicitly incorporating finite statistical fluctuations (Staffieri et al., 7 Jan 2026).

2. The FME Theorem: Finite-Size Key-Rate Bound

FME delivers explicit finite-size bounds in quantum cryptographic protocols. For generic QKD under collective attacks, if Alice and Bob share $n$ raw-key bits and Eve holds $E^n$, error correction reveals $\text{leak}_{\text{EC}}$ bits, and the total failure probability is $$\epsilon_{\text{sec}} = \epsilon_{\text{PE}} + \epsilon_{\text{EC}} + \epsilon_h$$ (parameter estimation, error correction, privacy amplification), then the key length $\ell$ satisfies (Staffieri et al., 7 Jan 2026):

$$\ell \leq H_{\min}^{\epsilon_s}(Z^n|E^n) - \text{leak}_{\text{EC}} + 2\log_2(2 \epsilon_h).$$

For collective-attack BB84,

$$P_{\text{guess}}(Z|E) = \tfrac12 + \sqrt{Q(1-Q)}, \qquad H_{\min}(Z|E) = -\log_2\left[\tfrac12 + \sqrt{Q(1-Q)}\right],$$

with $Q$ denoting the quantum bit error rate.

3. Methodology and Proof Techniques

The FME approach is algorithmic and implements quantifiable finite-size effects through:

• Parameter estimation: Statistical bounds (Hoeffding/Chernoff) determine confidence intervals for observed errors, e.g., choosing $$\delta = \sqrt{(1/(2m))\ln(1/\epsilon_{\text{PE}})}$$ for $m$ samples.
• Eve’s guessing probability: For i.i.d. rounds, $$P_{\text{guess}}(Z^n|E^n) = P_{\text{guess}}(Z|E)^n$$, maximized over admissible states.
• Privacy amplification: The quantum leftover-hash lemma relates smooth min-entropy to trace distance secrecy (e.g., $H_{\min}(Z^n|E^n) - 2\log_2(\sqrt{2}\epsilon_h)$).
• Error correction leakage: Explicit subtraction from the privacy amplification budget.
• Union bound: Aggregates all probability failure events.

These steps yield key rates resilient to block-size effects, remaining robust and nonzero even when other approaches (AEP, EUR) may fail (Staffieri et al., 7 Jan 2026).

4. Comparative Analysis: FME vs. EUR and AEP

A principal advantage of FME lies in its finite-size correction structure:

Bound Type	Correction Order	Block-Size Regime	Asymptotic Tightness	Finite-Size Rate Behavior
EUR	None	Large $N$	Optimal	Highest
AEP	$O(1/\sqrt{N})$	Moderate/Small $N$	Tight	May become pessimistic
FME	$O(1/N)$	Small $N$	Conservative	Positive when AEP fails

In BB84, FME delivers nonzero rates at blocks $N\sim5\times10^4$ when AEP vanishes, though EUR remains best if available. At higher QBER values, FME has a window where it is strictly superior to AEP. As $N\to\infty$, the asymptotic FME rate $$r_N^{\text{FME}}\to \eta(1-f)[-\log_2(\frac12+\sqrt{Q(1-Q)})-\gamma h_2(Q)]$$ approaches but remains below the EUR/AEP limiting rate (Staffieri et al., 7 Jan 2026).

5. Extensions: FME in Device Independence and Quantum Randomness

FME generalizes to device-independent protocols and quantum randomness generation via the Entropy Accumulation Theorem (EAT) (Mironowicz et al., 23 Jun 2025, Kamin et al., 2024). Here,

$$H_{\min}^{\epsilon_s}(A^nB^n|X^nY^nE) > n\,t - n(\epsilon_V+\epsilon_K) - \epsilon_\Omega,$$

with $t$ set by observed average min-tradeoff functions, and correction terms $\epsilon_V,O(\beta)$, $\epsilon_K,O(\beta^2)$, $\epsilon_\Omega,O(1/\beta)$, parametrized by protocol completion and statistical testing. SDP optimization yields affine min-tradeoff functions, incorporating device and Bell-type constraints, powering frameworks capable of returning numerical secure key rates or randomness output per second for large data sets. The block-size scaling of the correction terms critically controls protocol efficiency in practical scenarios.

6. Direct One-Shot Min-Entropy: Resource Perspective

Unsmooth FME captures exact one-shot capacities for private state transfer, quantum masking, and catalytic dephasing (Lie et al., 2020). For a density operator $\rho$,

$$H_{\min}(\rho) = -\log_2 \lVert \rho \rVert_\infty,$$

quantifies the largest dimension $d$ such that $\rho$ enables a one-shot protocol (private transfer, masking, dephasing) on a $d$-dimensional system iff $H_{\min}(\rho)\ge\log_2 d$. The capacity is strictly superadditive under resource tensor products. For security proofs in cryptography, this provides threshold conditionalities for resource states, independent of the asymptotic regime.

7. Computational and Practical Considerations

FME lends itself to efficient algorithmic estimation, both for quantum and classical scenarios. Min-entropy estimators based on Renyi-2 (collision entropy) or Coron’s test have closed-form or bisection solutions and offer low bias/high accuracy at moderate computational cost (Kim et al., 2020). For large cryptographic blocks or online applications, collision-entropy estimators enable lightweight implementations, although variance can increase for very-high-entropy cases. FME bounds drastically reduce required block lengths compared to traditional large-deviation analyses, rendering secure quantum protocols accessible in milliseconds of real-world data collection (Ng et al., 2012).

8. Applications and Protocol Scope

FME applies across discrete-variable QKD (BB84, six-state), device-independent QKD, decoy-state QKD, quantum random number generation (QRNG), and two-party primitives in noisy/bounded storage models. In continuous-variable settings, where tight entropic uncertainty relations are not available, FME provides tractable, positive bounds, underpinning privacy and correctness in composable security frameworks (Staffieri et al., 7 Jan 2026, Wang et al., 2024). FME’s adaptability to min-tradeoff optimization, source symmetrization, and eigenvalue-based one-shot lower bounds enables coverage of both finite- and infinite-dimensional quantum systems.

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For further protocol-specific derivations, implementation recipes, and numerical results, see (Staffieri et al., 7 Jan 2026, Mironowicz et al., 23 Jun 2025, Kamin et al., 2024, Lie et al., 2020, Wang et al., 2024, Ng et al., 2012, Kim et al., 2020).