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

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.

arXiv commentv1: 17 pages, comments welcome; v2: re-organize the structure of the paper

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• Quantum Causal Inference with Extremely Light Touch
• Certifying Quantum Temporal Correlation via Randomized Measurements: Theory and Experiment
• The spatiotemporal doubled density operator: a unified framework for analyzing spatial and temporal quantum processes
• Quantum dynamics as a pseudo-density matrix
• Multipartite quantum states over time from two fundamental assumptions
• Experimental Virtual Quantum Broadcasting
• Bipartite quantum states admitting a causal explanation