标题： 受约束的HRT表面及其熵解释 显示英文标题

标题： Constrained HRT Surfaces and their Entropic Interpretation

摘要： 考虑两个位于共同边界柯西超面上的边界子区域$A$和$B$，同时考虑与$B$相关的HRT表面$\gamma_B$。在这一背景下，约束HRT表面$\gamma_{A:B}$可以定义为锚定在$A$上的余维-2整体表面，该表面通过限制在包含$\gamma_B$的柯西片上的最大最小构造获得。 结果是，$\gamma_{A:B}$是两个部分$\gamma^B_{A:B}$和$\gamma^{\bar B}_{A:B}$的并集，分别位于$B$的纠缠楔形区域及其补集$\bar B$的纠缠楔形区域中。 与HRT表面$\gamma_A$的面积$\mathcal{A}\left(\gamma_A\right)$不同，在半经典极限下，$\gamma_{A:B}$的面积$\mathcal{A}\left(\gamma_{A:B}\right)$与$\gamma_B$的面积$\mathcal{A}\left(\gamma_B\right)$对易。 To study the entropic interpretation of $\mathcal{A}\left(\gamma_{A:B}\right)$, we analyze the Rényi entropies of subregion $A$ in a fixed-area state of subregion $B$. We use the gravitational path integral to show that the $n\approx1$ Rényi entropies are then computed by minimizing $\mathcal{A}\left(\gamma_A\right)$ over spacetimes defined by a boost angle conjugate to $\mathcal{A}\left(\gamma_B\right)$. 在块$\gamma^B_{A:B}$和$\gamma^{\bar B}_{A:B}$以恒定的提升角度相交的情况下，几何论证表明此时$n\approx1$瑞尼熵由$\frac{\mathcal{A}(\gamma_{A:B})}{4G}$给出。 我们讨论由于$n\to1$和$G\to0$极限的非对易性，$n\approx1$瑞尼熵与冯·诺依曼熵有何不同。 我们还讨论了行为如何随固定面积态的宽度变化。 我们的结果与尝试使用标准随机张量网络描述时间依赖几何的一些问题相关。 显示英文摘要

摘要： Consider two boundary subregions $A$ and $B$ that lie in a common boundary Cauchy surface, and consider also the associated HRT surface $\gamma_B$ for $B$. In that context, the constrained HRT surface $\gamma_{A:B}$ can be defined as the codimension-2 bulk surface anchored to $A$ that is obtained by a maximin construction restricted to Cauchy slices containing $\gamma_B$. As a result, $\gamma_{A:B}$ is the union of two pieces, $\gamma^B_{A:B}$ and $\gamma^{\bar B}_{A:B}$ lying respectively in the entanglement wedges of $B$ and its complement $\bar B$. Unlike the area $\mathcal{A}\left(\gamma_A\right)$ of the HRT surface $\gamma_A$, at least in the semiclassical limit, the area $\mathcal{A}\left(\gamma_{A:B}\right)$ of $\gamma_{A:B}$ commutes with the area $\mathcal{A}\left(\gamma_B\right)$ of $\gamma_B$. To study the entropic interpretation of $\mathcal{A}\left(\gamma_{A:B}\right)$, we analyze the R\'enyi entropies of subregion $A$ in a fixed-area state of subregion $B$. We use the gravitational path integral to show that the $n\approx1$ R\'enyi entropies are then computed by minimizing $\mathcal{A}\left(\gamma_A\right)$ over spacetimes defined by a boost angle conjugate to $\mathcal{A}\left(\gamma_B\right)$. In the case where the pieces $\gamma^B_{A:B}$ and $\gamma^{\bar B}_{A:B}$ intersect at a constant boost angle, a geometric argument shows that the $n\approx1$ R\'enyi entropy is then given by $\frac{\mathcal{A}(\gamma_{A:B})}{4G}$. We discuss how the $n\approx1$ R\'enyi entropy differs from the von Neumann entropy due to a lack of commutativity of the $n\to1$ and $G\to0$ limits. We also discuss how the behaviour changes as a function of the width of the fixed-area state. Our results are relevant to some of the issues associated with attempts to use standard random tensor networks to describe time dependent geometries.