Petz条件熵与夹层Rényi条件熵的严格连续性
Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies
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中文总结 AI 辅助
该研究确定了阶数在[1/2,1)区间的优化Petz与夹层Rényi条件熵的迹距离严格连续模，给出可达上界，验证其极限与已有量子条件熵连续界一致，通过线性化、Schmidt秩支配等方法完成证明。
中文摘要 AI 辅助
我们确定了对所有阶数$\alpha\in[\frac{1}{2},1)$的优化Petz条件熵和夹层Rényi条件熵在迹距离下的严格连续模。若两个两体态的迹距离不超过$\delta$，则两类条件熵的差值至多为$\frac{1}{1-\alpha}\log[(1-\varepsilon)^\alpha+(D-1)^{1-\alpha}\varepsilon^\alpha]$，其中$\varepsilon := \min\{\delta,1-1/D\}$，$D$为有效维度，由第一个子系统的维度乘以最大可能Schmidt秩给出。对每个$\delta\in[0,1]$的距离约束，该界都可由带最大纠缠锚态的各向同性态对达到。取$\alpha\uparrow1$可恢复Berta等人[arXiv:2607.24687]近期得到的量子条件熵严格连续界。证明过程在由各向同性等式族决定的比较点处对相关凹Rényi泛函进行线性化。Schmidt秩支配将等式几何推广到任意锚态，随后迹距离对偶性和非对易校准估计在不削弱严格常数的前提下控制了微扰项与锚态项。后一项估计需要矩阵分析，并借助了ChatGPT 5.6 Sol的辅助。
英文摘要
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.
