  Process tomography, the experimental characterization of physical processes,
is a central task in science and engineering. Here we investigate the axiomatic
requirements that guarantee the in-principle feasibility of process tomography
in general physical theories. Specifically, we explore the requirement that
process tomography should be achievable with a finite number of auxiliary
systems and with a finite number of input states. We show that this requirement
is satisfied in every theory equipped with universal extensions, that is,
correlated states from which all other correlations can be generated locally
with non-zero probability. We show that universal extensions are guaranteed to
exist in two cases: (1) theories permitting conclusive state teleportation, and
(2) theories satisfying three properties of Causality, Pure Product States, and
Purification. In case (2), the existence of universal extensions follows from a
symmetry property of Purification, whereby all pure bipartite states with the
same marginal on one system are locally interconvertible. Crucially, our
results hold even in theories that do not satisfy Local Tomography, the
property that the state of any composite system can be identified from the
correlations of local measurements. Summarizing, the existence of universal
extensions, without any additional requirement of Local Tomography, is a
sufficient guarantee for the characterizability of physical processes using a
finite number of auxiliary systems.
