We introduce and explore the notion of "spaces of input histories", a broad family of combinatorial objects which can be used to model input-dependent, dynamical causal order. We motivate our definition with reference to traditional partial order- and preorder-based notions of causal order, adopted by the majority of previous literature on the subject, and we proceed to explore the novel landscape of combinatorial complexity made available by our generalisation of those notions. In the process, we discover that the fine-grained structure of causality is significantly more complex than we might have previously believed: in the simplest case of binary inputs, the number of available "causally complete" spaces grows from 7 on 2 events, to 2644 on 3 events, to an unknown number on 4 events (likely around a billion). For perspective, previous literature on non-locality and contextuality used a single one of the 2644 available spaces on 3 events, work on definite causality used 19 spaces, derived from partial orders, and work on indefinite causality used only 6 more, for a grand total of 25. This paper is the first instalment in a trilogy: the sheaf-theoretic treatment of causal distributions is detailed in Part 2, "The Topology of Causality" [arXiv:2303.07148], while the polytopes formed by the associated empirical models are studied in Part 3, "The Geometry of Causality" [arXiv:2303.09017]. An exhaustive classification of the 2644 causally complete spaces on 3 events with binary inputs is provided in the supplementary work "Classification of causally complete spaces on 3 events with binary inputs", together with the algorithm used for the classification and partial results from the ongoing search on 4 events.
We introduce and explore the notion of "spaces of input histories", a broad family of combinatorial objects which can be used to model input-dependent, dynamical causal order. We motivate our definition with reference to traditional partial order- and preorder-based notions of causal order, adopted by the majority of previous literature on the subject, and we proceed to explore the novel landscape of combinatorial complexity made available by our generalisation of those notions. In the process, we discover that the fine-grained structure of causality is significantly more complex than we might have previously believed: in the simplest case of binary inputs, the number of available "causally complete" spaces grows from 7 on 2 events, to 2644 on 3 events, to an unknown number on 4 events (likely around a billion). For perspective, previous literature on non-locality and contextuality used a single one of the 2644 available spaces on 3 events, work on definite causality used 19 spaces, derived from partial orders, and work on indefinite causality used only 6 more, for a grand total of 25. This paper is the first instalment in a trilogy: the sheaf-theoretic treatment of causal distributions is detailed in Part 2, "The Topology of Causality" [arXiv:2303.07148], while the polytopes formed by the associated empirical models are studied in Part 3, "The Geometry of Causality" [arXiv:2303.09017]. An exhaustive classification of the 2644 causally complete spaces on 3 events with binary inputs is provided in the supplementary work "Classification of causally complete spaces on 3 events with binary inputs", together with the algorithm used for the classification and partial results from the ongoing search on 4 events.
We present an exhaustive classification of the 2644 causally complete spaces of input histories on 3 events with binary inputs, together with the algorithm used to find them. This paper forms the supplementary material for a trilogy of works: spaces of input histories, our dynamical generalisation of causal orders, are introduced in "The Combinatorics of Causality"; the sheaf-theoretic treatment of causal distributions is detailed in "The Topology of Causality"; the polytopes formed by the associated empirical models are studied in "The Geometry of Causality".
We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. Our work has its roots in the sheaf-theoretic framework for contextuality by Abramsky and Brandenburger, which it extends to include arbitrary causal orders (be they definite, dynamical or indefinite). We define a notion of causal function for arbitrary spaces of input histories, and we show that the explicit imposition of causal constraints on joint outputs is equivalent to the free assignment of local outputs to the tip events of input histories. We prove factorisation results for causal functions over parallel, sequential, and conditional sequential compositions of the underlying spaces. We prove that causality is equivalent to continuity with respect to the lowerset topology on the underlying spaces, and we show that partial causal functions defined on open sub-spaces can be bundled into a presheaf. In a striking departure from the Abramsky-Brandenburger setting, however, we show that causal functions fail, under certain circumstances, to form a sheaf. We define empirical models as compatible families in the presheaf of probability distributions on causal functions, for arbitrary open covers of the underlying space of input histories. We show the existence of causally-induced contextuality, a phenomenon arising when the causal constraints themselves become context-dependent, and we prove a no-go result for non-locality on total orders, both static and dynamical.
We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. We define causaltopes, our chosen portmanteau of "causal polytopes", for arbitrary spaces of input histories and arbitrary choices of input contexts. We show that causaltopes are obtained by slicing simpler polytopes of conditional probability distributions with a set of causality equations, which we fully characterise. We provide efficient linear programs to compute the maximal component of an empirical model supported by any given sub-causaltope, as well as the associated causal fraction. We introduce a notion of causal separability relative to arbitrary causal constraints. We provide efficient linear programs to compute the maximal causally separable component of an empirical model, and hence its causally separable fraction, as the component jointly supported by certain sub-causaltopes. We study causal fractions and causal separability for several novel examples, including a selection of quantum switches with entangled or contextual control. In the process, we demonstrate the existence of "causal contextuality", a phenomenon where causal inseparability is clearly correlated to, or even directly implied by, non-locality and contextuality.
The idea that events are equipped with a partial causal order is central to our understanding of physics in the tested regimes: given two pointlike events A and B , either A is in the causal past of B , B is in the causal past of A , or A and B are space-like separated. Operationally, the meaning of these order relations corresponds to constraints on the possible correlations between experiments performed in the vicinities of the respective events: if A is in the causal past of B , an experimenter at A could signal to an experimenter at B but not the other way around, while if A and B are space-like separated, no signaling is possible in either direction. In the context of a concrete physical theory, the correlations compatible with a given causal configuration may obey further constraints. For instance, space-like correlations in quantum mechanics arise from local measurements on joint quantum states, while time-like correlations are established via quantum channels. Similarly to other variables, however, the causal order of a set of events could be random, and little is understood about the constraints that causality implies in this case. A main difficulty concerns the fact that the order of events can now generally depend on the operations performed at the locations of these events, since, for instance, an operation at A could influence the order in which B and C occur in A ’s future. So far, no formal theory of causality compatible with such dynamical causal order has been developed. Apart from being of fundamental interest in the context of inferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order of events in quantum mechanics can be indefinite. Here, we develop such a theory in the general multipartite case. Starting from a background-independent definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal correlations. For a fixed number of settings and outcomes for each party, these correlations form a polytope whose facets define causal inequalities. The case of quantum correlations in this paradigm is captured by the process matrix formalism. We investigate the link between causality and the closely related notion of causal separability of quantum processes, which we here define rigorously in analogy with the link between Bell locality and separability of quantum states. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there are causally separable quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension. We characterize the class of ECS quantum processes in the tripartite case via simple conditions on the form of the process matrix. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.
Indefinite causal order (ICO) has been studied via higher-order quantum processes (e.g.\ the quantum switch), process matrices, and quantum-gravity proposals involving superposed causal structure, yet the meaning of ``indefiniteness'' and its relation to definite-order explanations often remain opaque.
Part~I develops a category-theoretic formulation of definite-order explainability as a gluing problem: each definite causal ordering (a partial order/DAG type) is treated as a context, and causal separability amounts to a consistent global section (possibly after convex mixing), whereas causal nonseparability is a failure-to-glue. We also introduce a compact seven-valued contextual classifier -- an intuitionistic elaboration -- that separates variation across contexts from genuine indeterminacy.
Part~II applies this framework to a quantum-gravity motivated setting where the fundamental time is a parametric ordering variable τ, distinct from geometric (spacetime) time. Adopting a stochastic-quantization perspective on spin-network dynamics (Hilbert space not assumed fundamental) and reading the Wheeler--DeWitt condition as an equilibrium/stationarity constraint, we interpret ICO as indeterminacy of the parametric order of coarse-grained relational interventions, even when the microscopic update process is globally ordered by τ. Together, the two parts provide a common language for comparing ICO criteria and for stating precisely what ``no hidden definite order'' means.
We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within categorical quantum mechanics, the objects of this theory encode fine-grained causal relationships between subsystems and give a new method for expressing and deriving consequences for a broad class of causal structures. We show that this framework enables one to define families of processes which are consistent with arbitrary acyclic causal orderings. In particular, one can define one-way signalling (a.k.a. semi-causal) processes, non-signalling processes, and quantum n-combs. Furthermore, our framework is general enough to accommodate recently-proposed generalisations of classical and quantum theory where processes only need to have a fixed causal ordering locally, but globally allow indefinite causal ordering. To illustrate this point, we show that certain processes of this kind, such as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner, and a classical three-party example due to Baumeler, Feix, and Wolf are all instances of a certain family of processes we refer to as SOCn in the appropriate category of higher-order causal processes. After defining these families of causal structures within our framework, we give derivations of their operational behaviour using simple, diagrammatic axioms. Comment: Extended version of a LICS 2017 paper with the same title