Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

• URL: http://arxiv.org/abs/2608.04947v1
• Date: Wed, 05 Aug 2026 15:15:19 GMT
• Title: Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies
• Abstract summary: 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.
• Score: 13.098901971644656
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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