3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Bloch sphere sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. Six common polarization types exist and are called Jones Vectors. Indeed Henri Poincaré was the first to suggest the use of this kind of geometrical representation at the end of 19th century, as a three-dimensional representation of Stokes parameters. The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space ℂ to the Bloch sphere is the Hopf fibration. Given an orthonormal basis, any pure state formula_3 of a two-level quantum system can be