  Predictions for measurement outcomes in physical theories are usually
computed by combining two distinct notions: a state, describing the physical
system, and an observable, describing the measurement which is performed. In
quantum theory, however, both notions are in some sense identical: outcome
probabilities are given by the overlap between two state vectors - quantum
theory is self-dual. In this paper, we show that this notion of self-duality
can be understood from a dynamical point of view. We prove that self-duality
follows from a computational primitive called bit symmetry: every logical bit
can be mapped to any other logical bit by a reversible transformation.
Specifically, we consider probabilistic theories more general than quantum
theory, and prove that every bit-symmetric theory must necessarily be
self-dual. We also show that bit symmetry yields stronger restrictions on the
set of allowed bipartite states than the no-signalling principle alone,
suggesting reversible time evolution as a possible reason for limitations of
non-locality.
