Quantum Causal Models

Abstract

It is known that the classical framework of causal models is not general enough to allow for causal reasoning about quantum systems. While the framework has been generalized in a variety of different ways to the quantum case, much of this work leaves open whether causal concepts are fundamental to quantum theory, or only find application at an emergent level of classical devices and measurement outcomes. Here, we present a framework of quantum causal models, with causal relations defined in terms intrinsic to quantum theory, and the central object of study being the quantum process itself. Following Allen et al., Phys. Rev. X 7, 031021 (2017), the approach defines quantum causal relations in terms of unitary evolution, in a way analogous to an approach to classical causal models that assumes underlying determinism and situates causal relations in functional dependences between variables. We show that any unitary quantum circuit has a causal structure corresponding to a directed acyclic graph, and that when marginalising over local noise sources, the resulting quantum process satisfies a Markov condition with respect to the graph. We also prove a converse to this statement. We introduce an intrinsically quantum notion that plays a role analogous to the conditional independence of classical variables, and (generalizing a central theorem of the classical framework) show that d-separation is sound and complete for it in the quantum case. We present generalizations of the three rules of the classical do-calculus, in each case relating a property of the causal structure to a formal property of the quantum process, and to an operational statement concerning the outcomes of interventions. In addition, we introduce and derive similar results for classical split-node causal models, which are more closely analogous to quantum causal models than the classical causal models that are usually studied.

arXiv comment61 pages, 28 figures, v2 has significantly expanded Introduction and Discussion, Section 4 rewritten for clarity, some citations added

References in corpus (39)

• Quantum correlations with no causal order
• Theoretical framework for quantum networks
• Negative entropy and information in quantum mechanics
• Probabilistic theories with purification
• The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Operational Markov condition for quantum processes
• Beyond Bell's Theorem: Correlation Scenarios
• Perfect discrimination of no-signalling channels via quantum superposition of causal structures
• Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication
• Quantum common causes and quantum causal models
• Information-Theoretic Implications of Quantum Causal Structures
• Semicausal operations are semilocalizable
• Quantum Dynamics as an analog of Conditional Probability
• Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios
• Causal categories: relativistically interacting processes
• On modifications of Reichenbach's principle of common cause in light of Bell's theorem
• Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity
• Nested Markov Properties for Acyclic Directed Mixed Graphs
• The space of logically consistent classical processes without causal order
• Operational quantum theory without predefined time
• Operational formulation of time reversal in quantum theory
• Pre- and post-selected quantum states: density matrices, tomography, and Kraus operators
• Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
• Quantum communication in a superposition of causal orders
• Markovian acyclic directed mixed graphs for discrete data
• A quantum causal discovery algorithm
• QBism: Quantum Theory as a Hero's Handbook
• Each instant of time a new Universe
• Which causal structures might support a quantum-classical gap?
• Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios
• Inability of the entropy vector method to certify nonclassicality in linelike causal structures
• Equivalence of relativistic causal structure and process terminality
• Quantum Causal Networks
• A time-reversible quantum causal model
• Factorization of Quantum Density Matrices According to Bayesian and Markov Networks
• Quantum causal models via QBism: the short version

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• A convergent inflation hierarchy for quantum causal structures
• Quantum Causal Unravelling
• Quantum enhancement to information acquisition speed
• A no-go theorem for superpositions of causal orders
• Implications of Local Friendliness violation for quantum causality
• Quantum causal inference in the presence of hidden common causes: An entropic approach
• Quantum Non-Gaussianity From An Indefinite Causal Order of Gaussian Operations