The notions of causal order and indefiniteness of causal order have popped up in information science. One can construct quantum computations, which cannot be carried out in fixed causal order: the simplest one is SWITCH which changes the causal order. It has been reported that the SWITCH has been realized experimentally with almost 7 sigma accuracy. In standard physics causal order is fixed and SWITCH is impossible so that the implications could be world view changing. In TGD Zero Energy Ontology (ZEO) replaces ordinary ontology and the arrow of time is not fixed, and it is interesting to see whether superposition of different causal orders related by time inversion T for causal diamond (CD) and SWITCH could be realized in ZEO. The twistor lift of TGD leads to the proposal that CD is accompanied by a Minkowskian generalization of self-dual Kahler form J(CD). Although the moduli space of CDs allows to avoid breaking of Poincare invariance, self-duality of J(CD) leads to violation of T implying that different causal orders correspond to disjoint sectors of world of classical worlds. This makes possible also superposition of different causal orders and SWITCH would map these sectors to each other.
The notions of causal order and indefiniteness of causal order have popped up in information science. One can construct quantum computations, which cannot be carried out in fixed causal order: the simplest one is SWITCH which changes the causal order. It has been reported that the SWITCH has been realized experimentally with almost 7 sigma accuracy. In standard physics causal order is fixed and SWITCH is impossible so that the implications could be world view changing. In TGD Zero Energy Ontology (ZEO) replaces ordinary ontology and the arrow of time is not fixed, and it is interesting to see whether superposition of different causal orders related by time inversion T for causal diamond (CD) and SWITCH could be realized in ZEO. The twistor lift of TGD leads to the proposal that CD is accompanied by a Minkowskian generalization of self-dual Kahler form J(CD). Although the moduli space of CDs allows to avoid breaking of Poincare invariance, self-duality of J(CD) leads to violation of T implying that different causal orders correspond to disjoint sectors of world of classical worlds. This makes possible also superposition of different causal orders and SWITCH would map these sectors to each other.
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.
Researchers have long been aiming to understand how the characteristics of Quantum Theory and General Relativity combine to account for regimes in their interface. One reason why this is a hard task is how differently the theories approach time and causality. For instance, causal structure in relativity is determined by the distribution of mass in spacetime while, in the quantum formalism, it is supposed to be fixed and given in advance. In this master's thesis, we discuss the notion of indefinite order, which first appears in an abstract generalization of Quantum Theory [...] where the demand for global causal structure is removed, in principle allowing cases for which the order of operations in protocols is not necessarily well defined. One epitomical example of indefinite order is the quantum switch process, which realizes a quantum superposition of orders of two operations on a target system. The quantum switch probabilities have been reproduced in experimental optical setups that are fully described in principle by quantum mechanics. Since these experiments are compatible with spacetime causal structure, this generated uncertainty about the conclusions that can be drawn from obtaining these results depending on the context. Here, we return to the initial motivations and also present how scenarios involving gravity in low energies could lead to indefinite order. This includes the formulation of a quantum switch in a quantum gravity scenario and of a quantum switch in a classical Schwarzschild metric. The switch then provides a common ground to discuss different kinds of setups. The latter proposal of a quantum switch in a classical metric is an original work that, aside from being an example of indefinite order, proposes the realization of the protocol in Earth's gravity as a test of quantum mechanics on curved spacetimes, a regime which has not yet been explored experimentally.
Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks—in particular, the violation of causal inequalities —which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computation—the quantum switch —is a genuine example of ‘indefinite causal order’. We do this by introducing a new tool—the causal witness —which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal inequality.
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.
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.
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.