Effective entropy of quantum fields coupled with gravity
Effective entropy of quantum fields coupled with gravity
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DOI：
10.1007/jhep10(2020)052
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发表时间：
2020-07
影响因子：
5.4
通讯作者：
Xi Dong;X. Qi;Shangnan Zhou;Zhenbin Yang
中科院分区：
文献类型：
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作者：
Xi Dong;X. Qi;Shangnan Zhou;Zhenbin Yang
纠缠熵，或冯诺依曼熵，量化了量子态的不确定性。对于弯曲空间中的量子场，对于固定的背景几何，量子场论自由度的纠缠熵是明确定义的。在本文中，我们提出了一个推广的量子场论纠缠熵，包括动力学引力。广义量有效熵及其Renyi熵的推广是通过副本计算的解析延拓定义的。复制理论被定义为在我们所研究的区域的边界上具有余维2膜的具有多个原始边界条件的引力路径积分。我们讨论了在规范不变条件下定义该区域的不同方法，并证明了有效熵满足量子极值面公式。当量子场具有较大的纠缠度时，量子极值曲面会发生拓扑跃迁，之后会出现纠缠岛区域。我们的结果将全息熵的Hubeny-Rangamani-Takayanagi公式（带量子修正）推广到没有渐近AdS边界的一般几何，并为解决诸如渐近平坦时空中蒸发黑洞的Page曲线等问题提供了一个更坚实的框架。我们应用公式的两个例子系统，一个封闭的二维宇宙和一个四维最大扩展Schwarzchild黑洞。我们讨论了随机张量网络模型中有效熵的模拟，它提供了对一般动力学几何中量子信息性质的更具体的理解。我们表明，在没有像AdS空间的情况下，一个大的边界，它是必要的，以正确地表征量子态和相关函数的随机张量网络中引入耦合到原始系统的附属。利用超密度算符理论，我们研究了带有辅助系统的纠缠岛，并给出了纠缠岛中的量子信息如何在依赖于态和依赖于量子态的映射中重构。我们研究了封闭宇宙（没有空间边界）的情况，并讨论了它是如何与开放宇宙。
Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.