This is an announcement for the paper "Dvoretzky's theorem and the complexity of entanglement detection" by Guillaume Aubrun and Stanislaw Szarek.
Abstract: The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbb{C}^d \otimes \mathbb{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathsf{I})(\rho)$ is not positive semi-definite. We show that that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on a study of the approximability of the set of states (resp. of separable states) by polytopes with few vertices or with few faces, and ultimately relies on the Dvoretzky--Milman theorem about the dimension of almost spherical sections of convex bodies. The result can be interpreted as a geometrical manifestation of the complexity of entanglement detection.
Archive classification: quant-ph math.FA
Mathematics Subject Classification: 81P40, 46B07
Submitted from: aubrun@math.univ-lyon1.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1510.00578
or
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alspach＠math.okstate.edu