We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.
We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.
Classically, the causal order of two timelike separated events A and B is fixed-either A before B or B before A. This is no longer true in quantum theory, where it is possible to encounter superpositions of causal orders. The quantum switch is one of the most prominent processes with indefinite causal order. Optical realizations of the quantum switch have been successfully implemented in experiments, but some argue that this merely simulates a process with indefinite causal order and that a superposition of spacetime metrics is required for a true realization. Here, we provide a relativistic definition of causal order between operationally defined events that defines a meaningful observable in both the general relativistic and quantum mechanical sense. Importantly, this observable does not distinguish between the indefinite causal order implemented on an optical bench and the gravitational quantum switch, a gedanken experiment where the indefinite causal order is achieved by a quantum superposition of gravitational fields. Therefore, our results support the thesis that the optical quantum switch is just as much a realization of indefinite causal order as its gravitational counterpart, which makes use of the quantum mechanical behavior of spacetime.
Investigating the role of causal order in quantum mechanics has recently revealed that the causal relations of events may not be a priori well defined in quantum theory. Although this has triggered a growing interest on the theoretical side, creating processes without a causal order is an experimental task. We report the first decisive demonstration of a process with an indefinite causal order. To do this, we quantify how incompatible our setup is with a definite causal order by measuring a "causal witness." This mathematical object incorporates a series of measurements that are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In our experiment, we perform a measurement in a superposition of causal orders-without destroying the coherence-to acquire information both inside and outside of a "causally nonordered process." Using this information, we experimentally determine a causal witness, demonstrating by almost 7 SDs that the experimentally implemented process does not have a definite causal order.
The process-matrix formalism describes quantum processes without assuming a fixed global causal order, but the physical meaning of indefinite causal order remains open. We address this question within histories theory, where temporal ordering and dynamical evolution are distinct structures. We show that the decoherence functional admits coherent superpositions of dynamics. In measurement settings, under suitable factorization conditions, these generate process matrices. Thus, we identify process-matrix indefinite causal order as an operationally restricted realization of the more general phenomenon of superposition of dynamics. Histories theory also admits a distinct kinematical notion of indefinite order, in which the ordering of physical events is itself a history observable. The resulting distinction between event order and intervention order clarifies the relation between indefinite causal order, quantum dynamics, and spacetime structure.
This work develops a graph-theoretic and process-based foundation for indefinite causal order in quantum theory, demonstrating that causal indefiniteness is not an ad hoc or exotic phenomenon but an emergent structural property of quantum dynamics when the primitive unit of evolution is taken to be the П-такт (P-Tact) — a discrete quantum process composed of local unitary superposition and entangling unitary interaction. Building upon a sequence of previously published foundational works, this paper formulates indefinite causal order as a superposition of quantum process graphs, where each graph encodes a definite causal structure, and the quantum state corresponds to a coherent superposition over such graphs. Within this framework, causal order is not fundamental but arises only in coarse-grained or classical limits. We show that: Indefinite causal order naturally emerges from the superposition of process graphs, without invoking external control systems or higher-order operations. The quantum switch is reinterpreted as a special finite realization of a more general graph-superposition principle. Time and causality emerge as effective parameters in the infinite-graph (continuum) limit of discrete П-такт dynamics. The graph-theoretic formulation is fully compatible with — and structurally supports — the Everettian (Many-Worlds) interpretation, where branching corresponds to graph bifurcations rather than wavefunction collapse. Classical spacetime causality appears as a macroscopic ordering imposed on an underlying non-causal quantum substrate. This approach unifies quantum information, indefinite causal order, emergent temporality, and Everettian branching within a single formal framework based on discrete process units and their graph composition. The results close several open conceptual gaps in the interpretation of quantum switches, process matrices, and non-classical causal structures, offering a consistent ontology grounded in quantum dynamics itself. This paper directly builds upon and extends the following foundational works: Pavlov, M. (2026). Quantum Switch as an Emergent Process Structure from Discrete Process Tacts (П-такт). Zenodo.https://doi.org/10.5281/zenodo.18498181 Pavlov, M. (2026). Quantum Switch and Teleportation. Zenodo.https://doi.org/10.5281/zenodo.18509173 Pavlov, M. (2026). Emergence of Causality, Gravity, Time, and Three-Dimensional Space from Discrete P-Tact (П-такт) Dynamics. Zenodo.https://doi.org/10.5281/zenodo.18504244 Pavlov, M. (2026). P-Tact (П-такт) as the Primitive Unit of Quantum Theory. Zenodo.https://doi.org/10.5281/zenodo.18516711 Together, these works establish a coherent program aimed at demonstrating the universality of quantum mechanics, the absence of fundamental collapse, and the emergence of spacetime and causality from discrete quantum process dynamics.
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 study the notion of causal orders for the cases of (classical and quantum) circuits and spacetime events. We show that every circuit can be immersed into a classical spacetime, preserving the compatibility between the two causal structures. Using the process matrix formalism, we analyse the realisations of the quantum switch using 4 and 3 spacetime events in classical spacetimes with fixed causal orders, and the realisation of a gravitational switch with only 2 spacetime events that features superpositions of different gravitational field configurations and their respective causal orders. We show that the current quantum switch experimental implementations do not feature superpositions of causal orders between spacetime events, and that these superpositions can only occur in the context of superposed gravitational fields. We also discuss a recently introduced operational notion of an event, which does allow for superpositions of respective causal orders in flat spacetime quantum switch implementations. We construct two observables that can distinguish between the quantum switch realisations in classical spacetimes, and gravitational switch implementations in superposed spacetimes. Finally, we discuss our results in the light of the modern relational approach to physics.