Quantum theory with indefinite causal structure

Quantum theory can be understood as a theory of information processing in the circuit framework for operational probabilistic theories. This approach presupposes a definite casual structure as well as a preferred time direction. But in general relativity, the causal structure of space-time is dynamical and not predefined, which indicates that a quantum theory that could incorporate gravity requires a more general operational paradigm. In this talk, I will describe recent progress in this direction. First, I will show how relaxing the assumption that local operations take place in a global causal structure leads to a generalized framework that unifies all signaling and non-signaling quantum correlations in space-time via an extension of the density matrix called the process matrix. This framework also contains a new kind of correlations incompatible with any definite causal structure, which violate causal inequalities, the general theory of which I am going to present. I will then present an extension of the process matrix framework, in which no predefined causal structure is assumed even locally. This is based on a more general, time-neutral notion of operation, which leads to new insights into the problem of time-reversal symmetry in quantum mechanics, the meaning of causality, and the fact that we remember the past but not the future. In the resultant generalized formulation of quantum theory, operations are associated with regions that can be connected in networks with no directionality assumed for the connections. The theory is compatible with timelike loops and other acausal structures.