Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

2026-08-05

AI Summary

This work establishes, for the first time, sharp continuity bounds for both Petz and sandwiched Rényi conditional entropies for all orders α ∈ [1/2, 1) in terms of trace distance. By linearizing the associated concave Rényi functionals around comparison points determined by families of isotropic identities, and combining Schmidt-rank dominance, trace-distance duality, and non-commutative perturbation estimates, the authors derive an optimal bound of the form (1/(1−α)) log[(1−ε)^α + (D−1)^{1−α} ε^α], where ε = min{δ, 1−1/D} and D denotes the effective dimension. This bound is tight for any trace-distance constraint and, in the limit α → 1, exactly recovers the known sharp continuity bound for quantum conditional entropy, thereby significantly advancing the theory of continuity for non-commutative entropies.

Abstract

We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order $α\in[\frac12,1)$. If two bipartite states are within trace distance $δ$, then both conditional entropies differ by at most $\frac{1}{1-α} \log[(1-\varepsilon)^α +(D-1)^{1-α}\varepsilon^α]$, where $\varepsilon := \min\{δ,1-1/D\}$ and $D$ is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint $δ\in[0,1]$, the bound is attained by an isotropic pair with a maximally entangled anchor. Taking $α\uparrow1$ recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave Rényi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.

Problem

• Rényi conditional entropy
• trace distance
• modulus of continuity
• quantum information
• entanglement

Innovation

• sharp continuity
• Rényi conditional entropy
• trace distance
• Schmidt rank
• noncommutative calibration