An elementary introduction to the geometry of quantum states with a picture book

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of n-qubits, the dimension is exponentially large in n. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra. When the dimension gets large there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.