  It is well-known that tensor decompositions show separations, that is, that
constraints on local terms (such as positivity) may entail an arbitrarily high
cost in their representation. Here we show that many of these separations
disappear in the approximate case. Specifically, for every approximation error
$\varepsilon$ and norm, we define the approximate rank as the minimum rank of
an element in the $\varepsilon$-ball with respect to that norm. For positive
semidefinite matrices, we show that the separations between rank, purification
rank, and separable rank disappear for a large class of Schatten $p$-norms. For
nonnegative tensors, we show that the separations between rank, positive
semidefinite rank, and nonnegative rank disappear for all $\ell_p$-norms with
$p>1$. For the trace norm ($p = 1$), we obtain upper bounds that depend on the
ambient dimension. We also provide a deterministic algorithm to obtain the
approximate decomposition attaining our bounds. Our main tool is an approximate
version of Carath\'eodory's Theorem. Our results imply that many separations
are not robust under small perturbations of the tensor, with implications in
quantum many-body systems and communication complexity.
