Geometry of quantum states : an introduction to quantum entanglement Convexity colours and statistics Geometry of probability distributions Much ado about spheres Complex projective spaces Outline of quantum mechanics Coherent states and group actions The stellar representation The space of density matrices Purification of mixed quantum states Quantum operations Duality : maps versus states Discrete structures in hilbert space Density matrices and entropies Distinguishability measures Monotone metrics and measures Quantum entanglement Multipartite entanglement Appendix A : basic notions of differential geometry