APS/O. Gühne and T. Moroder
The geometry of entanglement. In the high-dimensional space of all matrices, the set of all quantum states is convex (light blue), and the separable (nonentangled) states form a convex subset (darker blue). An entanglement witness W is a linear function that is positive on all separable states. In the figure, the line denotes the states in which W is zero, the state ρ1 is detected as entangled, while the state ρ2 is not detected by W. A possible nonlinear improvement N(W) of W may detect ρ1 and ρ2 at the same time, which is impossible with a single linear witness. Agnew et al.  have experimentally implemented a nonlinear witness that can achieve this in measurements of orbital angular momentum states of photons created in a nonlinear crystal.