有限资源下的纠缠容量与全息熵
The Capacity of Entanglement and Holographic Entropies at Finite Resources
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中文总结 AI 辅助
针对Ryu-Takayanagi公式对应关系的两个未决问题，研究引入纠缠容量给出无维度熵精度界，证明单个复制鞍点主导时极小曲面可确定所有单次熵，并界定了全息态偏离全息熵锥的程度。
中文摘要 AI 辅助
Ryu-Takayanagi公式将极小曲面的面积与边界子区域的冯·诺依曼熵等同起来，这一对应关系存在两个未解决的问题。第一个问题是几何对熵的确定精度有多高。Fannes-Audenaert不等式用希尔伯特空间维度给出了答案，但无论两个态有多接近，当截断被移除时，该维度都会发散。我们用纠缠容量（即模能量的方差）取而代之：纠缠容量的平方根随纠缠面积的平方根增长，而维度因子随正则化体积增长。该界是无维度的且可达，对于容量远小于$S_{vN}^2$乘以迹范数的任何微扰，它使得熵的不确定性呈次广延性。第二个问题是面积对单个态的意义是什么，因为压缩率和稀释率仅针对多副本定义，而几何描述的是单个态。当单个复制鞍点在$α=1$附近占主导时，所有固定$α>1$的光滑Rényi熵都与$S_{vN}$一致，误差为$\textit{O}(\text{√}S_{vN})$，光滑最小熵和最大熵也是如此。因此，极小曲面可同时确定该态的所有单次熵，其中大中心电荷扮演着渐近等分性质中大副本数的角色。由此我们界定了一个全息态可能看起来偏离全息熵锥的程度，而估计和认证问题仍有待解决。
英文摘要
The Ryu-Takayanagi formula equates the area of a minimal surface with the von Neumann entropy of a boundary subregion, and leaves two things about that identification open. The first is how sharply a geometry fixes an entropy. Fannes-Audenaert answers with a Hilbert-space dimension, which diverges as the cutoff is removed however close the two states are. We replace it with the capacity of entanglement, the variance of the modular energy, whose square root grows like the square root of the entangling area where the dimensional factor grows like the regulated volume. The bound is dimension-free and saturated, and it makes the ambiguity of the entropy subextensive for any perturbation whose capacity is small compared with $S_{vN}^2$ times the trace norm. The second is what the area means for a single state, since compression and dilution rates are defined only for many copies while a geometry describes one. When a single replica saddle dominates near $α= 1$, every smooth Rényi entropy at fixed $α> 1$ agrees with $S_{vN}$ to $\textit{O}(\sqrt{S_{vN}})$, as do the smooth min- and max-entropies. The minimal surface therefore fixes every one-shot entropy of the state at once, with large central charge playing the role of large copy number in the asymptotic equipartition property. As a consequence we bound how far outside the holographic entropy cone a holographic state can appear to fall, leaving estimation and certification open.
