Motivated by the corrected form of the entropy-area law, and with the help of von Neumann entropy of quantum matter, we construct an emergent spacetime by the virtue of the geometric language of statistical information manifolds. We discuss the link between Wald--Jacobson approaches of thermodynamic/gravity correspondence and Fisher pseudo-Riemannian metric of information manifold. We derive in detail Einstein's field equations in statistical information geometric forms. This results in finding a quantum origin of a positive cosmological constant that is founded on Fisher metric. This cosmological constant resembles those found in Lovelock's theories in a de Sitter background as a result of using the complex extension of spacetime and the Gaussian exponential families of probability distributions, and we find a time varying dynamical gravitational constant as a function of Fisher metric together with the corresponding Ryu-Takayanagi formula of such system. Consequently, we obtain a dynamical equation for the entropy in information manifold using Liouville-von Neumann equation from the Hamiltonian of the system. This Hamiltonian is suggested to be non-Hermitian, which corroborates the approaches that relate non-unitary conformal field theories to information manifolds. This provides some insights on resolving "the problem of time".

翻译：受熵-面积定律修正形式的启发，并借助量子物质的冯·诺伊曼熵，我们利用统计信息流形的几何语言构建了一个涌现时空。我们讨论了热力学/引力对应中的Wald-Jacobson方法与信息流形上的Fisher伪黎曼度量之间的联系。我们详细推导了以统计信息几何形式表述的爱因斯坦场方程。由此发现了一个基于Fisher度量的正宇宙学常数的量子起源。该宇宙学常数类似于在de Sitter背景下使用时空复延拓和高斯指数型概率分布族时Lovelock理论中得到的常数，并且我们得到了一个随Fisher度量变化的时变动力学引力常数，以及该体系对应的Ryu-Takayanagi公式。进而，我们利用系统的哈密顿量，通过Liouville-von Neumann方程得到了信息流形中熵的动力学方程。该哈密顿量被提议为非厄米形式，这佐证了将非幺正共形场论与信息流形联系起来的观点。这为解决"时间之矢问题"提供了一些见解。