# Tensor networks for quantum causal histories

## TL;DR

This paper develops tensor network representations for quantum causal histories, aiming to model states in quantum gravity and explore holographic mappings, with applications to quantum black holes and non-unitary evolutions.

## Contribution

It introduces a novel tensor network framework for quantum causal histories, utilizing channel-state duality and constructing matrix product and 2D tensor networks for quantum gravity models.

## Key findings

- Tensor networks can represent quantum causal histories with bipartite entangled states.
- Constructed tensor networks include matrix product states and 2D networks for entangled histories.
- Non-unitary local evolutions are essential for modeling bulk causal structures and black holes.

## Abstract

In this paper, we construct a tensor network representation of quantum causal histories, as a step towards directly representing states in quantum gravity via bulk tensor networks. Quantum causal histories are quantum extensions of causal sets in the sense that on each event in a causal set is assigned a Hilbert space of quantum states, and the local causal evolutions between events are modeled by completely positive and trace-preserving maps. Here we utilize the channel-state duality of completely positive and trace-preserving maps to transform the causal evolutions to bipartite entangled states. We construct the matrix product state for a single quantum causal history by projecting the obtained bipartite states onto the physical states on the events. We also construct the two dimensional tensor network states for entangled quantum causal histories in a restricted case with compatible causal orders. The possible holographic tensor networks are explored by mapping the quantum causal histories in a way analogous to the exact holographic mapping. The constructed tensor networks for quantum causal histories are exemplified by the non-unitary local time evolution moves in a quantum system on temporally varying discretizations, and these non-unitary evolution moves are shown to be necessary for defining a bulk causal structure and a quantum black hole. Finally, we comment on the limitations of the constructed tensor networks, and discuss some directions for further studies aiming at applications in quantum gravity.