Dvoretzky's Theorem and the Complexity of Entanglement Detection

TL;DR

This paper explores the geometric complexity of detecting entanglement in quantum states, showing that a super-exponential number of positive maps are needed to identify all robustly entangled states, using Dvoretzky's theorem.

Contribution

It establishes a connection between convex geometry and quantum entanglement detection, providing bounds on the number of positive maps required for robust entanglement detection.

Findings

Number of maps needed exceeds exp(c d^3 / log d)

Uses Dvoretzky's theorem to analyze convex bodies of quantum states

Reveals geometric complexity in entanglement detection

Abstract

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.