[论文解读] Black hole entropy from entanglement: A review

Saurya Das, S. Shankaranarayanan+1 more|arXiv (Cornell University)|Jun 2, 2008

Black Holes and Theoretical Physics参考文献 8被引用 33

一句话总结

本文提出黑洞熵源于标量场在事件视界两侧的量子纠缠，表明纠缠熵可重现基态或最小不确定性态下的贝肯斯坦-霍金面积律。对于激发态或叠加态，面积律出现幂律修正，其来源为远离视界的自由度贡献，且在视界面积较大时面积律得以恢复。

ABSTRACT

We review aspects of the thermodynamics of black holes and in particular take into account the fact that the quantum entanglement between the degrees of freedom of a scalar field, traced inside the event horizon, can be the origin of black hole entropy. The main reason behind such a plausibility is that the well-known Bekenstein-Hawking entropy-area proportionality -- the so-called `area law' of black hole physics -- holds for entanglement entropy as well, provided the scalar field is in its ground state, or in other minimum uncertainty states, such as a generic coherent state or squeezed state. However, when the field is either in an excited state or in a state which is a superposition of ground and excited states, a power-law correction to the area law is shown to exist. Such a correction term falls off with increasing area, so that eventually the area law is recovered for large enough horizon area. On ascertaining the location of the microscopic degrees of freedom that lead to the entanglement entropy of black holes, it is found that although the degrees of freedom close to the horizon contribute most to the total entropy, the contributions from those that are far from the horizon are more significant for excited/superposed states than for the ground state. Thus, the deviations from the area law for excited/superposed states may, in a way, be attributed to the far-away degrees of freedom. Finally, taking the scalar field (which is traced over) to be massive, we explore the changes on the area law due to the mass. Although most of our computations are done in flat space-time with a hypothetical spherical region, considered to be the analogue of the horizon, we show that our results hold as well in curved space-times representing static asymptotically flat spherical black holes with single horizon.

研究动机与目标

• 探究标量场在视界处的量子纠缠是否可解释黑洞熵。
• 确定纠缠熵重现贝肯斯坦-霍金面积律的条件。
• 探索当标量场处于激发态或叠加量子态时，面积律的修正形式。
• 识别贡献于纠缠熵的自由度在空间中的分布，尤其在非基态情形下。
• 考察标量场质量与弯曲时空几何对纠缠熵及面积律的影响。

提出的方法

• 在弯曲时空中的标量场进行正则量子化，分解为球谐函数与径向模态。
• 通过对内部区域（r ≤ r_h）进行约化，将外部视为子系统，计算约化密度矩阵。
• 通过约化密度矩阵的冯·诺依曼熵计算纠缠熵。
• 采用勒梅特坐标以避免视界处的坐标奇点，并简化哈密顿量形式。
• 执行正则变换，将场哈密顿量映射为平坦空间中自由标量场的哈密顿量，从而实现解析处理。
• 分析纠缠熵对量子态（基态、相干态、压缩态、激发态或叠加态）及场质量的依赖性。

实验结果

研究问题

• RQ1 标量场的纠缠熵能否重现黑洞熵的贝肯斯坦-霍金面积律？
• RQ2 当标量场处于激发态或叠加量子态时，面积律会引入何种修正？
• RQ3 纠缠熵的贡献在视界及其周围的空间分布如何？
• RQ4 标量场质量在修正纠缠熵的面积律中起何作用？
• RQ5 在平坦时空中以截断面为模型的结果，能在多大程度上推广至具有单视界的弯曲时空黑洞？

主要发现

• 标量场在基态或最小不确定性态（如相干态或压缩态）下，其纠缠熵重现了贝肯斯坦-霍金面积律。
• 当标量场处于激发态或基态与激发态的叠加态时，面积律出现幂律修正，其大小随视界面积A的增大而趋于∼1/A。
• 在非基态情形下，远离视界的自由度对纠缠熵有显著贡献，这与基态中近视界模态占主导地位的情况不同。
• 随着视界面积增大，幂律修正逐渐减弱，且在大面积极限下面积律被渐近恢复。
• 引入质量标量场会改变纠缠熵，但面积律及其修正的定性行为保持一致。
• 在平坦时空中以假设的球形截断面（类比于视界）推导出的结果，可推广至具有单视界的静态、球对称黑洞（包括史瓦西黑洞）的弯曲时空情形。