Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson’s bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order. Quantum theory has been shown to be compatible with processes violating a definite causal order, but no detailed analysis of the maximum allowed degree of violation has been carried out so far. Here, the authors fill this gap by providing the analogue of Tsirelson’s bound for the violation of causal inequalities.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson’s bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order. Quantum theory has been shown to be compatible with processes violating a definite causal order, but no detailed analysis of the maximum allowed degree of violation has been carried out so far. Here, the authors fill this gap by providing the analogue of Tsirelson’s bound for the violation of causal inequalities.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order.
Quantum theory allows for the superposition of causal orders between operations, i.e., for an indefinite causal order; an implication of the principle of quantum superposition. Since a higher theory might also admit this feature, an understanding of superposition and indefinite causal order in a generalised probabilistic framework is needed. We present a possible notion of superposition for such a framework and show that in maximal theories, respecting non-signalling relations, single system state-spaces do not admit superposition; however, composite systems do. Additionally, we show that superposition does not imply entanglement. Next, we provide a concrete example of a maximally Bell-nonlocal theory, which not only admits the presented notion of superposition, but also allows for post-quantum violations of theory-independent inequalities that certify indefinite causal order; even up to an algebraic bound. These findings might point towards potential connections between a theory's ability to admit indefinite causal order, Bell-nonlocal correlations and the structure of its state spaces.
The study of causal relations has recently been applied to the quantum realm, leading to the discovery that not all physical processes have a definite causal structure. While indefinite causal processes have previously been experimentally shown, these proofs relied on the quantum description of the experiments. Yet, the same experimental data could also be compatible with definite causal structures within different descriptions. Here, we present the first demonstration of indefinite temporal order outside of quantum formalism. We show that our experimental outcomes are incompatible with a class of generalised probabilistic theories satisfying the assumptions of locality and definite temporal order. To this end, we derive physical constraints (in the form of a Bell-like inequality) on experimental outcomes within such a class of theories. We then experimentally invalidate these theories by violating the inequality using entangled temporal order. This provides experimental evidence that there exist correlations in nature which are incompatible with the assumptions of locality and definite temporal order.
An astonishing feature of higher-order quantum theory is that it can accommodate indefinite causal order. In the simplest bipartite setting, there exist signaling correlations for which it is fundamentally impossible to ascribe a definite causal order for the parties' actions. Moreover, the assumptions required to arrive at such a statement (local quantum transformations and well-behaved probabilities) result in a nontrivial set of correlations, whose boundary is, to date, uncharacterized. In this work, we investigate indefinite causal order in boxworld theories. We construct a higher-order theory whose descriptor is the generalized bit (gbit) - the natural successor of the classical bit and quantum qubit. By fixing the local transformations in boxworld and asking about the global causal structure, we find that we trivially recover the full set of two-way signaling correlations. In light of this, we motivate and propose two physical principles in order to limit the set of achievable correlations: nonsignaling preservation and no signaling without system exchange. We find that a higher-order boxworld theory that respects these physical principles leads to (i) a nontrivial set of achievable correlations and (ii) a violation of some causal inequalities that is higher than what can be achieved in higher-order quantum theory. These results lead us to conjecture that the set of correlations of our higher-order boxworld theory is an outer approximation to the set of correlations produced by higher-order quantum theory.
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 no-signalling principle lies at the heart of Bell-type experiments involving spacelike-separated parties. Although not all no-signalling correlations can be realized within quantum theory, the no-sinalling principle itself remains fundamental even for post-quantum theories, since its violation would imply superluminal signalling. In the spirit of device independent nature of the Bell inequalities, there has recently been growing interest in causal inequalities, which identify correlations admitting no causal explanation. Such correlations have been rigorously studied within the process-matrix formalism, which is built upon a constraint analogous to spacelike separation in Bell scenarios. We refer to this constraint as the single-interaction constraint, whereby each party interacts with the environment only once. Despite considerable effort, no fundamental guiding principle is currently known that characterizes the restrictions this constraint imposes on the correlations generated by Process Matrices. In this work, we address this gap by proposing a generalized causality principle for the bipartite case. We show that for certain cases, the generalised causality principle defines a strict subset of the set of all signalling correlations. This principle thus provides a fundamental limit on quantum and even post-quantum theories subject to the single-interaction constraint. By analogy with no-signalling, whose violation excludes certain causal structures among the parties, a violation of the generalised causality principle rules out a corresponding class of causal structures(causal structures with the single interaction constraint). More broadly, our techniques provide a general framework for deriving generalised causal constraints, applicable to a wide class of indefinite-causal-order scenarios.