Quantum Operations with Indefinite Time Direction

The fundamental dynamics of quantum particles is neutral with respect to the arrow of time. And yet, our experiments are not: we observe quantum systems evolving from the past to the future, but not the other way round. A fundamental question is whether it is possible, at least in principle, to conceive a broader set of operations that probe quantum processes in the backward direction, from the future to the past, or more generally, in a combination of the forward and backward directions. Here we introduce a mathematical framework for operations that are not constrained to a definite time direction. More generally, we introduce a set of multipartite operations that include indefinite time direction as well as indefinite causal order, providing a framework for potential extensions of quantum theory.