Causal loops escape paradoxes if they generate valid probabilities under arbitrary independent local interventions. We show that a deterministic classical process is logically consistent if and only if it contains no signaling loops: cyclic signaling is forbidden but cyclic causation is not, allowing indefinite causal order. Logical consistency is then found to hold exactly when the list of events of a process is pairwise exclusive and complete, providing the first event-based, intervention-free, non-recursive characterization of consistent causal loops. Our results repair a previously introduced intervention-free criterion that misses global loops and reveal that what makes deterministic classical communications without causal order consistent is a multipartite form of Specker's principle: if any two of several questions can be answered jointly, so can all of them.
Causal loops escape paradoxes if they generate valid probabilities under arbitrary independent local interventions. We show that a deterministic classical process is logically consistent if and only if it contains no signaling loops: cyclic signaling is forbidden but cyclic causation is not, allowing indefinite causal order. Logical consistency is then found to hold exactly when the list of events of a process is pairwise exclusive and complete, providing the first event-based, intervention-free, non-recursive characterization of consistent causal loops. Our results repair a previously introduced intervention-free criterion that misses global loops and reveal that what makes deterministic classical communications without causal order consistent is a multipartite form of Specker's principle: if any two of several questions can be answered jointly, so can all of them.
Bell non-local correlations cannot be naturally explained in a fixed causal structure. This serves as a motivation for considering models where no global assumption is made beyond logical consistency. The assumption of a fixed causal order between a set of parties, together with free randomness, implies device-independent inequalities—just as the assumption of locality does. It is known that local validity of quantum theory is consistent with violating such inequalities. Moreover, for three parties or more, even the (stronger) assumption of local classical probability theory plus logical consistency allows for violating causal inequalities. Here, we show that a classical environment (with which the parties interact), possibly containing loops, is logically consistent if and only if whatever the involved parties do, there is exactly one fixed-point, the latter being representable as a mixture of deterministic fixed-points. We further show that the non-causal view allows for a model of computation strictly more powerful than computation in a world of fixed causal orders.
Quantum processes can exhibit indefinite causal order, allowing coherent superpositions of distinct orderings of events. Whether such processes generate genuinely noncausal statistics, however, depends not only on their global causal structure but also on the local operations available to the parties. Here, we restrict communication to the classical outcomes of local operations, yielding a causal polytope with new causal inequalities that can be violated using measure-and-reprepare, entanglement-breaking instruments requiring no coherent transmission through the laboratories. Using a semidefinite-programming framework, we bound the achievable violations and reveal noncausal correlations that satisfy all conventional input-signalling inequalities, demonstrating that restricting the information available for communication can uncover forms of indefinite causal order inaccessible under less restrictive operational assumptions.
The fundamental nature of causality has been challenged by quantum theory. In particular, recent developments have shown that by relaxing the standard assumption that quantum events are embedded in a fixed background causal structure, a new kind of causal relations, indefinite, can emerge. These indefinite causal structures have been interpreted as causal loops allowing for the violation of a form of causality without paradoxes. On the other hand, quantum contextuality shows that any attempt to extend quantum theory such that classical logic holds in the extended domain fails due to the rise of inconsistent logical loops. There has been increasing evidence that it is a key feature in the understanding quantum paradoxes. This thesis aims at improving our understanding of indefinite causal orders and quantum paradoxes in order to address the question: “Could the non-fundamental nature of causality be related to a form of contextuality ?”The process matrix formalism is a mathematical framework which, in analogy withentanglement and the violation of Bell inequalities (nonlocality), offers new tools - witnesses of causal nonseparability and the violation of causal inequalities with noncausalcorrelations - to study and identify indefinite causal orders. In a bottom-up approach, wehave identified a large class of physically implementable processes, the quantum circuitswith quantum control of causal order (QC-QCs), that includes causally nonseparable processes such as the celebrated “quantum switch”. However these circuits cannot be used to violate causal inequalities. Nevertheless, in another work, I have proposed a significant relaxation of assumptions to certify the causal nonseparability of many processes, showing that it can be done in a semi-device-independent scenario with untrusted local operations and trusted quantum inputs. In particular, I have shown that the quantum switch can generate noncausal correlations in this kind of scenario.In order to refine my understanding of quantum contextuality, I have investigated theorigins of the Kochen-Specker theorem and I have analysed how the Hardy paradox entails a form of logical contextuality. I have identified that the two possible intermediatemeasurements involved in a pre-post-selection version of the paradox differ causally fromeach other. This analysis might be useful to study of the Frauchiger-Renner paradox, a“meta-version” of the Hardy paradox. On a more metaphysical aspect, I have argued fora “Gödelian hunch” from quantum theory, the idea that quantum paradoxes emerge froma lack of distinction between theoretical and metatheoretical objects.Finally, in order to establish a link between indefinite causal orders and contextuality,I have proposed a new causal game, in which the grandfather paradox is formalized interms of a logical inequality. Some process matrices can violate the causal inequalitywithout violating the logical one. I suggest that the emergence of logical inconsistenciesfrom the violation of a logical inequality might be interpreted as a form of contextuality.
Bell scenarios are multipartite scenarios that exclude any signalling between parties. This leads to a strict hierarchy of classical, quantum, and non-signalling correlations in such scenarios. Here we consider a minimal relaxation of non-signalling: each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. Crucially, unlike Bell, we make no global assumption about causal relations between parties, e.g., they could be embedded in some exotic spacetime with indefinite causal order. We do make a causal assumption local to each party, i.e., the input received by it causally precedes the output it sends out. We then ask: Can we device-independently certify the nonclassicality of multipartite correlations in such scenarios, just as Bell inequality violations do so in Bell scenarios? A priori, this is not clear: without some assumptions on the underlying physics (e.g., non-signalling), parties can realize arbitrary correlations. We therefore make a minimal assumption of logical consistency on the underlying physics, i.e., it must be free of time-travel antinomies without imposing any restrictions on local interventions of the parties. We then define antinomicity as a device-independent notion of nonclassicality and prove a strict hierarchy between correlation sets based on their antinomicity. An antinomic correlation cannot be explained by a classical physical theory compatible with free local interventions on pain of logical contradictions in the theory. On the other hand, parties exchanging quantum systems can witness antinomicity while respecting logical consistency. Antinomicity reduces to Bell nonlocality for non-signalling parties. It also resolves a conceptual puzzle, namely, the failure of causal inequalities as witnesses of nonclassicality: antinomicity implies causal inequality violations, but not conversely.
Causality is fundamental to science, but it appears in several different forms. One is relativistic causality, which is tied to a spacetime structure and forbids signaling outside the future. A second is an operational notion of causation that considers the flow of information between physical systems and interventions on them. In [V. Vilasini and R. Colbeck, General framework for cyclic and fine-tuned causal models and their compatibility with space-time, Phys. Rev. A 106, 032204 (2022).PLRAAN2469-992610.1103/PhysRevA.106.032204], we propose a framework for characterizing when a causal model can coexist with relativistic principles such as no superluminal signaling, while allowing for cyclic and nonclassical causal influences and the possibility of causation without signaling. In a theory without superluminal causation, both superluminal signaling and causal loops are not possible in Minkowski spacetime. Here we demonstrate that if we only forbid superluminal signaling, superluminal causation remains possible and show the mathematical possibility of causal loops that can be embedded in a Minkowski spacetime without leading to superluminal signaling. The existence of such loops in the given spacetime could in principle be operationally verified using interventions. This establishes that the physical principle of no superluminal signaling is not by itself sufficient to rule out causal loops between Minkowski spacetime events. Interestingly, the conditions required to rule out causal loops in a spacetime depend on the dimension. Whether such loops are possible in three spatial dimensions remains an important open question.
By relating and ordering events, causality constitutes a pivotal feature of our world. On the one hand, there are information-theoretic notions of causality defined in terms of the information processing ability of agents and on the other hand, there are relativistic notions of causality tied to a spacetime. In this thesis, we improve upon a framework introduced by V. Vilasini and R. Colbeck in PRA, 106, 032204 (2022) and PRL, 129, 110401 (2022) for connecting these notions, where the possibility of operationally detectable causal loops embedded in (1+1)-Minkowski spacetime without superluminal signalling was demonstrated. In the first part, we take the information-theoretic point of view, where the concept of higher-order (HO) affects relations was proposed to generically model signalling in the presence of cyclic, fine-tuned and non-classical causal influences. We establish new properties of HO affects relations and apply them to infer causal structures. We then demonstrate a complete and constructive way to detect causal loops from a set of HO affects relations. In the second part, we study the embedding of information-theoretic causal structures into partially ordered spacetimes. We propose stability conditions on the spacetime embedding to rule out a class of operationally detectable loops that cannot be ruled out by the principle of no-signalling (outside the relativistic future) alone. We then propose novel order-theoretic properties that we conjecture to hold in Minkowski spacetime with d≥2 spatial dimensions. This would imply that in contrast to (1+1)-Minkowski spacetime, in higher dimensions, the no-signalling principle is indeed sufficient for ruling out this class of loops. Finally, we introduce a relation which allows for causal inference through knowledge about the absence of signalling relations and we relate it to novel order-theoretic features of spacetime.