Classical correlations without predefined causal order arise from processes where parties manipulate random variables, and where the order of these interactions is not predefined. No assumption on the causal order of the parties is made, but the processes are restricted to be logically consistent under any choice of the parties’ operations. It is known that for three parties or more, this set of processes is larger than the set of processes achievable in a predefined ordering of the parties. Here, we model all classical processes without predefined causal order geometrically and find that the set of such processes forms a polytope. Additionally, we model a smaller polytope—the deterministic-extrema polytope—where all extremal points represent deterministic processes. This polytope excludes probabilistic processes that must be—quite unnaturally— fine-tuned , because any variation of the weights in a decomposition into deterministic processes leads to a logical inconsistency.
Classical correlations without predefined causal order arise from processes where parties manipulate random variables, and where the order of these interactions is not predefined. No assumption on the causal order of the parties is made, but the processes are restricted to be logically consistent under any choice of the parties’ operations. It is known that for three parties or more, this set of processes is larger than the set of processes achievable in a predefined ordering of the parties. Here, we model all classical processes without predefined causal order geometrically and find that the set of such processes forms a polytope. Additionally, we model a smaller polytope—the deterministic-extrema polytope—where all extremal points represent deterministic processes. This polytope excludes probabilistic processes that must be—quite unnaturally— fine-tuned , because any variation of the weights in a decomposition into deterministic processes leads to a logical inconsistency.
Quantum theory in a global spacetime gives rise to nonlocal correlations, which cannot be explained causally in a satisfactory way; this motivates the study of theories with reduced global assumptions. Oreshkov, Costa, and Brukner [Nat. Commun. 3, 1092 (2012)] proposed a framework in which quantum theory is valid locally but where, at the same time, no global spacetime, i.e., predefined causal order, is assumed beyond the absence of logical paradoxes. It was shown for the two-party case, however, that a global causal order always emerges in the classical limit. Quite naturally, it has been conjectured that the same also holds in the multiparty setting. We show that, counter to this belief, classical correlations locally compatible with classical probability theory exist that allow for deterministic signaling between three or more parties incompatible with any predefined causal order.
We consider the most general correlations that can be obtained by a group of parties whose causal relations are well defined, although possibly probabilistic and dependent on past parties' operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.
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.
The idea that events are equipped with a partial causal order is central to our understanding of physics in the tested regimes: given two pointlike events A and B , either A is in the causal past of B , B is in the causal past of A , or A and B are space-like separated. Operationally, the meaning of these order relations corresponds to constraints on the possible correlations between experiments performed in the vicinities of the respective events: if A is in the causal past of B , an experimenter at A could signal to an experimenter at B but not the other way around, while if A and B are space-like separated, no signaling is possible in either direction. In the context of a concrete physical theory, the correlations compatible with a given causal configuration may obey further constraints. For instance, space-like correlations in quantum mechanics arise from local measurements on joint quantum states, while time-like correlations are established via quantum channels. Similarly to other variables, however, the causal order of a set of events could be random, and little is understood about the constraints that causality implies in this case. A main difficulty concerns the fact that the order of events can now generally depend on the operations performed at the locations of these events, since, for instance, an operation at A could influence the order in which B and C occur in A ’s future. So far, no formal theory of causality compatible with such dynamical causal order has been developed. Apart from being of fundamental interest in the context of inferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order of events in quantum mechanics can be indefinite. Here, we develop such a theory in the general multipartite case. Starting from a background-independent definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal correlations. For a fixed number of settings and outcomes for each party, these correlations form a polytope whose facets define causal inequalities. The case of quantum correlations in this paradigm is captured by the process matrix formalism. We investigate the link between causality and the closely related notion of causal separability of quantum processes, which we here define rigorously in analogy with the link between Bell locality and separability of quantum states. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there are causally separable quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension. We characterize the class of ECS quantum processes in the tripartite case via simple conditions on the form of the process matrix. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.
In a scenario where two parties share, act on and exchange some physical resource, the assumption that the parties' actions are ordered according to a definite causal structure yields constraints on the possible correlations that can be established. We show that the set of correlations that are compatible with a definite causal order forms a polytope, whose facets define causal inequalities . We fully characterize this causal polytope in the simplest case of bipartite correlations with binary inputs and outputs. We find two families of nonequivalent causal inequalities; both can be violated in the recently introduced framework of process matrices , which extends the standard quantum formalism by relaxing the implicit assumption of a fixed causal structure. Our work paves the way to a more systematic investigation of causal inequalities in a theory-independent way, and of their violation within the framework of process matrices.
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.