# Connecting processes with indefinite causal order and multi-time quantum states

## TL;DR

This paper establishes a novel connection between process matrices and multi-time quantum states, offering new insights into indefinite causal order and its experimental realization.

## Contribution

It demonstrates the equivalence of process matrices to certain pre- and post-selected quantum states, providing a new conceptual framework and experimental approach.

## Key findings

- Process matrices are equivalent to a class of multi-time quantum states.
- Provides an explicit probabilistic recipe for experimental implementation.
- Shows indefinite causal order can exist with definite temporal order.

## Abstract

Recently, the possible existence of quantum processes with indefinite causal order has been extensively discussed, in particular using the formalism of process matrices. Here we give a new perspective on this question, by establishing a direct connection to the theory of multi-time quantum states. Specifically, we show that process matrices are equivalent to a particular class of pre- and post- selected quantum states. This offers a new conceptual point of view to the nature of process matrices. Our results also provide an explicit recipe to experimentally implement any process matrix in a probabilistic way, and allow us to generalize some of the previously known properties of process matrices. Furthermore we raise the issue of the difference between the notions of indefinite temporal order and indefinite causal order, and show that one can have indefinite causal order even with definite temporal order.