Dvoretzky's Theorem and the Complexity of Entanglement Detection

The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathrm{Id})(\rho)$ is not positive semi-definite. We show 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 the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.