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. CONTENTS I. Introduction 1 II. Quantum space-time causality and pseudo-density operator formalism 2 A. Pseudo-density operator 3 B. Quasi-probabilistic mixture of space-time product states 5 C. Space-time purification 6 III. Pseudo density operator marginal problem 6 A. Space-time separable marginal problem 7 B. Space-time symmetric extension 8 C. Polygamy of space-time correlations 8 D. Classical quasi-probability marginal problem 8 IV. Inferring global space-time state from reduced space-time states 9 A. Entropy of space-time states 9 B. Space-time maximum entropy principle 11 C. The neural network approach to inferring the global space-time state 12 V. Conclusion and discussion 12 Acknowledgments 13A. Quantum pseudo-channel 13 1. Quantum pseudo-channel as higher-order maps 13

Unification of spatiotemporal quantum formalisms: mapping between process and pseudo-density matrices via multiple-time states

“…Therefore it treats temporal correlations the same way it treats spatial correlations, integrating them into a unified framework. The PDM has been employed to explore various aspects of temporal quantum processes [15,27,[30][31][32][33][34][35], including quantum causal inference [15,36], quantum communication [32], temporal quantum teleportation [35], and compatibility of quantum processes [34], among others.…”