Approximate tensor decompositions: Disappearance of many separations

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