  The phenomenon of data hiding, i.e. the existence of pairs of states of a
bipartite system that are perfectly distinguishable via general entangled
measurements yet almost indistinguishable under LOCC, is a distinctive
signature of nonclassicality. The relevant figure of merit is the maximal ratio
(called data hiding ratio) between the distinguishability norms associated with
the two sets of measurements we are comparing, typically all measurements vs
LOCC protocols. For a bipartite $n\times n$ quantum system, it is known that
the data hiding ratio scales as $n$, i.e. the square root of the real dimension
of the local state space of density matrices. We show that for bipartite
$n_A\times n_B$ systems the maximum data hiding ratio against LOCC protocols is
$\Theta\left(\min\{n_A,n_B\}\right)$. This scaling is better than the
previously obtained upper bounds $O\left(\sqrt{n_A n_B}\right)$ and
$\min\{n_A^2, n_B^2\}$, and moreover our intuitive argument yields constants
close to optimal. In this paper, we investigate data hiding in the more general
context of general probabilistic theories (GPTs), an axiomatic framework for
physical theories encompassing only the most basic requirements about the
predictive power of the theory. The main result of the paper is the
determination of the maximal data hiding ratio obtainable in an arbitrary GPT,
which is shown to scale linearly in the minimum of the local dimensions. We
exhibit an explicit model achieving this bound up to additive constants,
finding that the quantum mechanical data hiding ratio is only of the order of
the square root of the maximal one. Our proof rests crucially on an unexpected
link between data hiding and the theory of projective and injective tensor
products of Banach spaces. Finally, we develop a body of techniques to compute
data hiding ratios for a variety of restricted classes of GPTs that support
further symmetries.
