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It is well-known that tensor decompositions show separations, that is, thatconstraints on local terms (such as positivity) may entail an arbitrarily highcost in their representation. Here we show that many of these separationsdisappear in the approximate case. Specifically, for every approximation error$\varepsilon$ and norm, we define the approximate rank as the minimum rank ofan element in the $\varepsilon$-ball with respect to that norm. For positivesemidefinite matrices, we show that the separations between rank, purificationrank, and separable rank disappear for a large class of Schatten $p$-norms. Fornonnegative tensors, we show that the separations between rank, positivesemidefinite 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 theambient dimension. We also provide a deterministic algorithm to obtain theapproximate decomposition attaining our bounds. Our main tool is an approximateversion of Carath\'eodory's Theorem. Our results imply that many separationsare not robust under small perturbations of the tensor, with implications inquantum many-body systems and communication complexity.

Approximate tensor decompositions: disappearance of many separations