We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $δ$ and $d=\dim A$ , the optimal dimension-only modulus of continuity is $h_2(δ)+δ\log(d^2-1)$ up to $δ=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$ , this bound is tight for every $δ\in[0,1]$ . The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $δ$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(δ)+δ\log(d^2-1)$ up to $δ=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$, this bound is tight for every $δ\in[0,1]$. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.

quant-pharXiv:2607.24687Jul 28, 20260 citations

Optimal uniform continuity bound for conditional entropy of classical–quantum states

Mark M. Wilde

arXiv:1909.01755Quantum Information ProcessingJan 2, 202015 citations

Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

Hao-Chung Cheng, Po-Chieh Liu

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.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.

quant-pharXiv:2608.04947Aug 6, 20260 citations

On the continuity of entropy

Gisela A. Lassner, Gisela A. Lassner

Reports on Mathematical PhysicsFeb 1, 19796 citations

Bounds for smooth min- and max-entropy

Yuan Zhai, Bo Yang, Zhengjun Xi

Quantum Information ProcessingMay 19, 20231 citation

We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most

δ

and

d=\dim A

, the optimal dimension-only modulus of continuity is

h_2(δ)+δ\log(d^2-1)

up to

δ=1-d^{-2}

and

2\log d

thereafter, where

h_2

denotes the binary entropy. When

\dim B\ge d

, this bound is tight for every

δ\in[0,1]

. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.

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.