Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

2026/08/05 by Hao-Chung Cheng, Po-Chieh Liu
Physics and Astronomy · Computer Science · Mathematics · #quant-ph #cs.IT #math-ph #math.IT #math.MP

paper · pdf

Relevant literature: arXiv:2007.05049, arXiv:2607.24687

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order α∈[\frac12,1). If two bipartite states are within trace distance δ, then both conditional entropies differ by at most (1)/(1-α) log[(1-ε)^α +(D-1)^1-αε^α], where ε := 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∈[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.

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