Figure 1:
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 ρ_{1} is detected as entangled, while the state ρ_{2} is not detected by W. A possible nonlinear improvement N(W) of W may detect ρ_{1} and ρ_{2} at the same time, which is impossible with a single linear witness. Agnew et al. [1] have experimentally implemented a nonlinear witness that can achieve this in measurements of orbital angular momentum states of photons created in a nonlinear crystal.