Dvoretzky's Theorem and the Complexity of 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.

Accepted talk in QIP'16. V2: added a variant of the main theorem showing that very highly entangled states are already hard to detect. V3: published version; improved presentation, results unchanged

References in corpus (5)

• A complete family of separability criteria
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• A quasipolynomial-time algorithm for the quantum separability problem
• Limitations of semidefinite programs for separable states and entangled games
• There is no direct generalization of positive partial transpose criterion to the three-by-three case

Cited by in corpus (3)

• Random matrix techniques in quantum information theory
• Limitations of semidefinite programs for separable states and entangled games
• On Approximations of the PSD Cone by a Polynomial Number of Smaller-sized PSD Cones