# An elementary introduction to the geometry of quantum states with pictures

## TL;DR

This paper provides an elementary, visual introduction to the geometry of quantum states, illustrating high-dimensional convex bodies and their properties, including separability and entanglement, using simple geometric shapes and pictures.

## Contribution

It offers a visual and elementary approach to understanding the complex geometry of quantum states, emphasizing cross sections and high-dimensional shapes.

## Key findings

- Quantum state space resembles a high-dimensional ball in most directions.
- The space of states is well-approximated by simple geometric shapes like balls and simplices.
- Geometric properties of entangled and separable states are discussed.

## Abstract

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.