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Measuring unconventional causal structures in monitored dynamics

Causality underpins all logical reasoning. However, the causal structure in quantum processes can be far from intuitive, often differing from its classical counterpart in relativity, which is defined by the light cone. In particular, in systems with measurement and post-selection, causal influence can occur between spacelike separated regions. In this work, we study the causal structure and emergent "arrow of time" in monitored quantum dynamics, particularly their dependence on initial and final states. We propose a new measure, the cross-entropy quantum causal influence, to quantify the extent of causal influence, whose simulation demonstrates exotic causal structures, such as inverted light cones. This quantity can be measured in current quantum computing platforms. Additionally, we provide an analytical understanding of the relation between time arrow and entropy by studying two types of models that are analytically tractable: a quantum Brownian evolution model and a dual-unitary circuit model.