Sharp Continuity of Petz and Sandwiched R\'enyi Conditional Entropies

Abstract

We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched R\'enyi conditional entropies for every order $\alpha\in[\frac12,1)$. If two bipartite states are within trace distance $\delta$, then both conditional entropies differ by at most $\frac{1}{1-\alpha} \log[(1-\varepsilon)^{\alpha} +(D-1)^{1-\alpha}\varepsilon^{\alpha}]$, where $\varepsilon := \min\{\delta,1-1/D\}$ and $D$ is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint $\delta\in[0,1]$, the bound is attained by an isotropic pair with a maximally entangled anchor. Taking $\alpha\uparrow1$ recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave R\'enyi 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.