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Bloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system named after the physicist Felix Bloch. Alternatively, it is the pure state space of a 1 qubit quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and pure states can be explicitly given. In generalized form, the Bloch sphere may also refer to the analogous space of an n-level quantum system. Quantum mechanics is mathematically formulated in Hilbert space or Projective Hilbert space. The space of pure states of a quantum system is given by the rays in the Hilbert space (the "points" of projective Hilbert space). The space of rays in any vector space is a projective space, and in particular, the space of rays in a two dimensional Hilbert space is the complex projective line, which is isomorphic to a sphere. Each pair of antipodal points on the Bloch sphere corresponds to a mutually exclusive pair of states of the particle, namely, spin up and spin down for a Stern-Gerlach experiment oriented along a particular axis in physical space. The natural metric on the Bloch sphere is the Fubini-Study metric. * 1 The qubit * 2 A generalization for pure states * 3 The geometry of density operators |