Quantum space-time marginal problem: global causal structure from local causal information

TL;DR

This paper investigates how to infer the global causal structure of space-time from local quantum causal information using pseudo-density operators, proposing a maximum entropy method and exploring quantum pseudo-channels.

Contribution

It introduces a space-time marginal problem framework, demonstrates solution existence, and develops a neural network-based method for determining global causal structures.

Findings

Almost always a solution exists for the space-time marginal problem.

The maximum entropy principle can effectively determine global causal structures.

Quantum pseudo-channel marginal problems can be transformed into pseudo-density operator problems.

Abstract

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators, and quantum causality between the involved events in an experiment is encoded in the corresponding pseudo-density operator. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the pseudo-density operators; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle, which can be solved effectively by using a neural network. The notion of quantum pseudo-channel is also introduced and we demonstrate that the quantum pseudo-channel marginal problem can be solved by transforming it into a pseudo-density operator marginal problem via the channel-state duality.