  In this paper, we explore the non-Hermitian transition matrix and its gravity
dual. States in quantum field theories or gravity theories are typically
prepared using Euclidean path integrals. We demonstrate that it is both natural
and necessary to introduce non-Hermitian transitions to describe the state when
employing different inner products in Euclidean quantum field theories.
Transition matrices that are $\eta$-pseudo-Hermitian, with $\eta$ being
positive-definite, play the same role as density matrices, where the operator
$\eta$ is closely related to the definition of the inner product. Moreover,
there exists a one-to-one correspondence between these transition matrices and
density matrices. In the context of AdS/CFT correspondence, the Euclidean path
integral in the boundary field theory can be translated to the bulk
gravitational path integral. We provide an overview of the construction and
interpretation of non-Hermitian spacetime. Specifically, we demonstrate the
crucial role of the non-Hermitian transition matrix in realizing the
thermofield concept in general cases and in understanding the gravity states
dual to the eternal black hole. In this context, the pseudoentropy of the
transition matrix can also be interpreted as black hole entropy. Finally, we
highlight the strong subadditivity property of pseudoentropy, and the
connection between non-Hermitian transition matrices and complex metrics.
