arXiv 1809.035162018-09-10hep-th

纠缠平台上模哈密顿量的性质

Properties of Modular Hamiltonians on Entanglement Plateaux

Raimond Abt, Johanna Erdmenger

DOI：: 10.1007/JHEP11(2018)002

摘要：

约化态的模块化哈密顿量，基本上由约化密度矩阵的对数给出，鉴于其与量子信息的关系，在AdS/CFT对应中起着重要的作用。特别是，它是状态之间距离的量子信息度量的基本成分，例如相对熵和Fisher信息度量。然而，模哈密顿量仅在少数例子中明确已知。对于由标量$λ$参数化的态族$ρ_λ$，通过纠缠第一定律，模块哈密顿量对$ρ_λ$和参考态$ρ_{λ_0}$之间的相对熵的$\tildeλ=λ-λ_0$中的一阶贡献完全由纠缠熵确定。对于几个示例，例如，对于CFTs的基态中的球形区域，已知高阶贡献消失。在这些情况下，模块化的哈密顿有助于费舍尔信息度量在一个平凡的方式。我们调查在哪些条件下的模块化的哈密顿提供了一个非平凡的贡献的Fisher信息度量，即当模块化的哈密顿的贡献相对熵是高阶的$\tildeλ$。我们考虑了两个纠缠区上的单参数约化态族，这两个纠缠区形成一个纠缠平台，即两个纠缠区的纠缠熵使Araki-Lieb不等式饱和.我们表明，在一般情况下，至少有一个相对熵的两个纠缠区域预计将涉及$\tildeλ$贡献的高阶模块化哈密顿量。此外，我们认为这一观察的影响突出的AdS/CFT的例子，形成纠缠高原在大$N$限制。

英文摘要：

The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states $ρ_λ$ that is parametrized by a scalar $λ$, the first order contribution in $\tildeλ=λ-λ_0$ of the modular Hamiltonian to the relative entropy between $ρ_λ$ and a reference state $ρ_{λ_0}$ is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in $\tildeλ$. We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve $\tildeλ$ contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large $N$ limit.

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