Abstract: 
We provide a proposal generalizing the asymptotic Euclidean state preparation to a state preparation by the deformation of a partial Cauchy slice in Lorentzian spacetime. With a proper choice of boundary conditions, we claim that the state forms a Hilbert space $\bf{H}_\sigma$ associated to the boundary $\sigma$ of the partial Cauchy slice. 

Assuming that $\bf{H}_\sigma$ factorises over connected components of $\sigma$, our framework admits operators and partial traces. This allows us to explore the information-theoretic structure of the states we define. As an example, we consider negative cosmological spacetime, and study a family of partial Cauchy slices of a BTZ black hole, where $\sigma$ consists of two circles. We consider the reduced state on one circle, and we find that its R\’enyi entropies are positive and monotonic, as required. Moreover, the von Neumann entropy is given not by the minimal area on the preparation slice, but by the Bekenstein-Hawking entropy of the black hole.

Time permitting, I will also present some thoughts of applying similar techniques to positive cosmological constant cases in the end.