Abstract
Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes – provided they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent.
Highlights
Causal reasoning is essential to science, yet quantum theory challenges it
The most general evolution of a system, assuming that it is initially uncorrelated with its environment, is given by a completely positive trace-preserving (CPTP) map E : LðHAÞ ! LðHBÞ, where this notation allows the output system to be different from the input system
The operation corresponds to a quantum instrument, which is a collection of LðHBÞg, such cthomat pEle1⁄4telyPkpEoksiitsivae (CP) maps fEk : LðHAÞ trace-preserving CP map
Summary
Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. 34,35 is that causal relations between quantum systems, as encoded in a DAG, correspond to influence through underlying unitary transformations This facilitated, in particular, a justification of the quantum Markov condition relative to a DAG that underpins the definition of a quantum causal model—any such model can be thought of as arising from a unitary circuit fragment with a compatible causal structure by marginalizing over latent local disturbances[35]. It is a natural question whether these hitherto separate lines of research can be merged to arrive at a causal model perspective on processes that are not compatible with a fixed order of the quantum nodes. While this direction of thought has been considered in earlier work (see, e.g., refs. 45,49,50), it was previously not clear how to take the idea forward due to various conceptual and technical obstacles—including, for example, how quantum nodes and the quantum Markov condition should be defined, how the notion of the autonomy of causal mechanisms should be understood[49], and how to prevent paradoxes
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