  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 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 these
operations, showing that some of them cannot be interpreted as random mixtures
of operations that probe processes in a definite direction. As a concrete
example, we construct an operation, called the quantum time flip, that probes
an unknown dynamics in a quantum superposition of the forward and backward
directions. This operation exhibits an information-theoretic advantage over all
operations with definite direction. It can be realised probabilistically using
quantum teleportation, and can be reproduced experimentally with photonic
systems. 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.
