We present an operational and model-independent framework to investigate the concept of no-backwards-in-time signalling. We define no-backwards-in-time signalling conditions, closely related to the spatial no-signalling conditions. These allow for theoretical possibilities in which the future affects the past, nevertheless without signalling backwards in time. This is analogous to non-local but no-signalling spatial correlations. Furthermore, our results shed new light on situations with indefinite causal structure and their connection to quantum theory.
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 general relativity, `causal structure' refers to the partial order on space-time points (or regions) that encodes time-like relationships. Recently, quantum information and quantum foundations saw the emergence of a `causality principle'. In the form used in this paper, which we call `process terminality', it states that when the output of a process is discarded, then the process itself may as well be discarded. While causal structure and process terminality at first seem to be entirely different notions, they become equivalent when making explicit what the partial order actually encodes, that is, that an event in the past can influence one in its future, but not vice-versa. The framework in which we establish this equivalence is that of process theories. We show how several previous results are instances of this result and comment on how this framework could provide an ideal, minimal canvas for crafting theories of quantum gravity.
Quantum mechanics is traditionally formulated using the reversible coordinate time
t, which enforces a globally reversible unitary group structure and time‑symmetric propagators. This conflicts with physically irreversible processes such as decoherence, measurement, entropy production, causal propagation, and curvature‑driven focusing in general relativity. We resolve this mismatch by replacing
t with the monotonic temporal field
T, previously developed in the context of global causal structure, temporal focusing, and worldline ordering in curved spacetime. Once
T is taken as the fundamental evolution parameter, quantum dynamics cannot be reversible: the inverse operator
U−1 is not physically realizable, and unitary evolution becomes a forward‑directed semigroup. We formulate **forward unitarity**, in which physical evolution is generated only by forward‑directed operators
Uphys(T2,T1) with
T2>T1. Pure states evolve according to a forward Schrödinger equation with an intrinsic, monotonic decoherence functional
D(T), while mixed states evolve under a forward Lindblad structure. Probability is conserved, but coherent recoverability is forbidden by geometric irreversibility, causal structure, and temporal focusing. We construct a Forward‑QM Lagrangian, a forward Heisenberg picture, and a forward path integral restricted to histories with non‑decreasing
T, showing how operator algebras, propagators, correlation functions, and creation/annihilation operators inherit the arrow of time. Forward‑QM provides a unified, forward‑directed formulation of quantum mechanics compatible with general relativity, causal geometry, and the irreversible structure of spacetime. It replaces reversible unitary groups with physically meaningful semigroup evolution, aligns quantum dynamics with measurement, decoherence, and entropy production, and establishes the foundation for Forward Quantum Field Theory (FQFT). The extension to FQFT—covering forward propagators, interacting fields, renormalization in the presence of monotonic decoherence, and causal compatibility in curved spacetime—will be developed in future work.
In the book the idea of irreversibility as an inherent property of time is developed theoretically and experimentally. The matter is related with causality, and the method of causal analysis is presented. The quantum causal analysis helps understand the principle of week causality which admits extraction of information from the future without the classical paradoxes. It implies a possibility of observation of the future as the existing reality. So, the acceptance of time irreversibility leads to a striking manifestation of reversibility – signaling in reverse time. Quantum insight allows considering correlations of the distant irreversible processes as nonlocal ones originated from a macroscopic entanglement. The experimental approach to study of macroscopic nonlocality is discussed, and design of the experimental setup is described. The results of experiments on macroscopic nonlocal correlations, the signals in reverse time and their application to the forecast of large-scale random processes are expounded.
Gell-Mann and Hartle (GMH) have recently considered time-neutral cosmological models in which the initial and final conditions are independently specified, and several authors have investigated experimental tests of such models. We point out here that GMH time-neutral models can allow superluminal signaling, in the sense that it can be possible for observers in those cosmologies, by detecting and exploiting regularities in the final state, to construct devices which send and receive signals between space-like separated points. In suitable cosmologies, any single superluminal message can be transmitted with probability arbitrarily close to one by the use of redundant signals. However, the outcome probabilities of quantum measurements generally depend on precisely which past and future measurements take place. As the transmission of any signal relies on quantum measurements, its transmission probability is similarly context dependent. As a result, the standard superluminal signaling paradoxes do not apply. Despite their unusual features, the models are internally consistent. These results illustrate an interesting conceptual point. The standard view of Minkowski causality is not an absolutely indispensable part of the mathematical formalism of relativistic quantum theory. It is contingent on the empirical observation that naturally occurring ensembles can be naturally pre-selected but not post-selected.