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 and 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，并借助量子物质冯·诺依曼熵的辅助，我们通过统计信息流形的几何语言构建了一个涌现时空。探讨了Wald与Jacobson热力学/引力对应方法同信息流形菲舍尔伪黎曼度量之间的联系，详细推导出以统计信息几何形式表达的爱因斯坦场方程。这一过程揭示了基于菲舍尔度量的正宇宙学常数的量子起源——该宇宙学常数类似于洛弗洛克理论在德西特背景下的结果，源于时空复延拓与高斯指数型概率分布族的应用。同时发现一个随时间变化的动力学引力常数，它作为菲舍尔度量的函数，与相应系统的Ryu-Takayanagi公式共同存在。最终，利用系统哈密顿量的刘维尔-冯·诺依曼方程，推导出信息流形中熵的动力学方程。该哈密顿量被建议为非厄米形式，佐证了将非幺正共形场论与信息流形相联系的研究路径，为理解"时间问题"提供了新视角。