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.
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.
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.
This paper is the thirty-fourth in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of independently existing objects. It is the capstone (third panel) of the ninth grouping — the quantum-mechanics–gravity interface trilogy (32 Quantum Geometry → 33 Records Backreacting on Geometry → 34 Causal Structure Without a Global Order) — and closes that trilogy. Its core physics is the process-matrix framework (Oreshkov, Costa, & Brukner, 2012), with the quantum switch (Chiribella et al., 2013) as the worked instance and causal (non)separability as the central classification. When the order of operations is quantum-controlled — and, at the interface, when geometry is superposed so different branches carry different causal graphs — the temptation is to posit a hidden definite global total order behind the indefiniteness. The thesis is that there is no Global Causal Sequencer; P1 is a declared architectural commitment, not a theorem, and a rival that posits a deeper definite order underwriting the indefiniteness is acknowledged but declined architecturally. R-A: indefinite operational order is not causal inconsistency or a closed loop — a coherent superposition of order components is causally nonseparable (witness-certifiable; note that witness-based nonseparability and causal-inequality violation are not equivalent, and the quantum switch need not violate a causal inequality) yet a valid, consistent process. R-B (restricted to controlled-order processes such as the quantum switch, or any process with a declared physically recorded order-control observable): recording the order-control does not by itself produce a determinate order — one must distinguish the coherent control, the decohered-but-unconditioned causally separable mixture, and the conditioned record-relative effective order — only the last is determinate, and only relative to a recorded value and the declared parameter chi, whose order-control basis is an admissible decoherence-stable pointer basis/algebra (not an arbitrary rotation). R-C: causal structure is profile-valued, CausalProf = (d_order, d_interv, d_record, d_valid, d_family, d_arrow), with a process-validity (not graph-acyclicity) component and an independent record-arrow component from Paper 9's arrow package. R-D (the novelty witness): effective order is provenance-gradable — a matched pair of histories with identical audited world-side order records but different provenance yields different warranted standing (OrderStandProf). R-E: a conservative-before-radical discipline works indefinite operational order first, then a minimal superposed-causal-graph bridge on Paper 32's identity fibers, separating an identity disposition from an order disposition; full quantization of the causal relation is flagged, not claimed. This retires the Global Causal Sequencer and closes the interface trilogy; no causal loops, no retrocausal signaling, no revived global order, no quantum-gravity completion. Presents no new physics; results are interpretive (L2) on imported physics (L1).
In quantum mechanics, quantum nonseparability is at the core of philosophical debates regarding its meaning. In the context of the process matrix formalism, causal nonseparability characterises quantum processes (connecting the inputs and outputs of different local quantum operations) that are incompatible with any definite causal structure among interacting parties. One talks about indefinite causal orders. A famous example of causally nonseparable processes is called the quantum switch (QS). It is extensively studied in the literature in virtue of its simple architecture and its various implementations in laboratories.The present work will discuss the possible interpretations of the QS's causal nonseparability under the assumption that causal nonseparability is seen as pointing towards novel objective features of nature.It was first defended that a scientific realist approach towards the process matrix formalism, which is an operational theory generalising quantum mechanics, was as much legitimate as any possible antirealist reading, contrary to certain views found in the literature. The reason is that operational formalisms are ontologically and epistemically neutral.From there, the theoretical concepts of interest, namely causal nonseparability and its model-independent counterpart called noncausality, were analysed in more details, in order to highlight in what sense they are distinct from the standard notions of quantum nonseparability and nonlocality in quantum mechanics.The discussion then focused on noncausality. It was argued that noncausality has a strong connection with a notion of temporal nonlocality. In the same way that Bell nonlocality is given different underlying explanations depending on the details of the chosen quantum mechanics' account, noncausality pointing towards temporal nonlocality can be given a variety of underlying descriptions depending on the exact way to interpret the process matrix formalism.The last chapter focused precisely on this particular point, namely on the various ways to understand process matrices and causal nonseparability in a realist context. In order to explore the potential impact of causal nonseparability on spacetime,we shifted from the notion of (indefinite) causal structure to (indefinite) spatiotemporal ones. This shift was allowed under a set of reasonable assumptions regarding the properties of a physical spacetime manifold and the connection between an operational and relativistic notions of causal relations.While different readings were suggested for indefiniteness of spatiotemporal relations, we insisted in particular on an objective understanding appealing to the concept of metaphysical indeterminacy. It was argued that such an approach could prove useful in a more general theoretical context such as quantum gravity, while being already partly supported in standard quantum mechanics.
In recent years, the investigation of causal relations in quantum theory has attracted a lot of interest. In particular, it has been found that it is possible to conceive of scenarios where some parties perform operations that are compatible with quantum theory locally, but that cannot be embedded into a global background causal structure. Such indefinite causal structures are of interest from a fundamental point of view, but also from the perspective of quantum information processing, since they do not fit into the usual paradigm of quantum circuits, which assumes a definite causal order. The main aim of this thesis is to study indefinite quantum causal relations involving more than two parties. Compared to the bipartite case, there are many new aspects and complications that arise in multipartite situations, which need to be clarified in order to fundamentally understand quantum causal structures, and to shed light on their implications and potential usefulness for quantum information processing.A suitable mathematical framework for the investigation of quantum causal relations is the process matrix formalism. In this framework, the notion of causal nonseparability was introduced in order to qualify scenarios that are incompatible with a definite causal order. In Chapter 2, we study how to generalise this concept to the multipartite case, how to characterise multipartite causally (non)separable quantum processes, and how to certify multipartite causal nonseparability. Another important topic is to determine which quantum scenarios with indefinite causal order are physically implementable, and how they can be realised concretely. In Chapter 3, we introduce two new classes of physically realisable multipartite quantum processes, and characterise them in terms of their process matrix descriptions. In particular, we define the class of quantum circuits with quantum control of causal order. The simplest example of such a circuit is the quantum switch, a protocol in which the order between two operations is controlled by a qubit in a superposition state, and which defines a causally nonseparable process. The class we introduce also contains more general examples of causally nonseparable processes with new features. We then show how the process matrix characterisation of this class allows us to search for new quantum information processing applications of causal nonseparability. In Chapter 4, we investigate a particular quantum communication effect in a scenario involving coherent control between two quantum channels. This leads us to a more general analysis of the notion of a coherently controlled channel, which involves certain subtilities. In Chapter 5, we turn to another problem that arises in multipartite scenarios, namely whether a given phenomenon is genuinely multipartite or not. More particularly, we study genuinely multipartite (non)causal correlations. In Chapter 6, we show that anomalous weak values are possible without post-selection. In Chapter 7, we show that certain tripartite process matrices, namely those that are unitarily extensible, have a realisation on so-called time-delocalised subsystems, i.e., quantum subsystems that are not associated with a definite time. The class of unitarily extensible tripartite process matrices is larger than the class of quantum circuits with quantum control, and in particular contains processes that violate so-called causal inequalities.
We propose a theoretical framework in which time is not a fundamental variable but an emergentproperty arising from the transition between indefinite and definite causal order in quantum processes.Building on the process matrix formalism and the established phenomenon of indefinite causal order,we introduce the concept of Pre-Time, a fully quantum domain where no global temporal directionexists and causal relations remain in coherent superposition. Within this framework, we define acausal-emergence measure λ(W ) that quantifies the distance between a given process matrix W andthe convex set of causally separable processes; λ(W ) > 0 identifies genuinely pre-temporal systems,whereas λ(W ) = 0 corresponds to the emergence of a classical temporal order. We show that, underincreasing decoherence or environmental coupling, any physically realizable process must undergo amonotonic reduction of λ(W ), providing a mathematical mechanism for the conformation of temporalorder. Within this framework, apparent retrocausal paradoxes associated with delayed-choice andquantum eraser experiments do not arise, since the “past” is not a pre-existing entity but a structureformed only upon causal consolidation. The model remains fully consistent with quantum mechanics,relativity, and thermodynamic irreversibility, while offering novel predictions about intermediatepre-temporal phases and the emergence of classical causality. Our results suggest that time ariseswhen the universe selects a definite causal order, and that the pre-temporal quantum regime may playa central role in foundational physics and future quantum-gravity research.
One of the most fundamental open problems in physics is the unification of general relativity and quantum theory to a theory of quantum gravity. An aspect that might become relevant in such a theory is that the dynamical nature of causal structure present in general relativity displays quantum uncertainty. This may lead to a phenomenon known as indefinite or quantum causal structure, as captured by the process matrix framework. Due to the generality of that framework, however, for many process matrices there is no clear physical interpretation. A popular approach towards a quantum theory of gravity is the Page-Wootters formalism, which associates to time a Hilbert space structure similar to spatial position. By explicitly introducing a quantum clock, it allows to describe time-evolution of systems via correlations between this clock and said systems encoded in history states. In this paper we combine the process matrix framework with a generalization of the Page-Wootters formalism in which one considers several observers, each with their own discrete quantum clock. We describe how to extract process matrices from scenarios involving such observers with quantum clocks, and analyze their properties. The description via a history state with multiple clocks imposes constraints on the physical implementation of process matrices and on the perspectives of the observers as described via causal reference frames. While it allows for describing scenarios where different definite causal orders are coherently controlled, we explain why certain non-causal processes might not be implementable within this setting.