Geometry of Quantum States

Abstract

Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d-dimensional system form a convex set in the d^2 - 1 dimensional Euclidean space (R^d^2) - 1 . For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S^2 of this unit sphere consists of all the pure states of the system. Every unitary transformation U an element of SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case, the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R^8 is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S^7 in R^8 will correspond to a state of the system. So not every rotation in R^8 will correspond to a valid unitary transformation of the 3 dimensional Hilbert Space. This gives a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, the author studies its two dimensional and three dimensional cross sections. The discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems are described in particular. This group turns out to be the same as that of a tetrahedron.