Editing Bloch sphere (section)

{{short description|Geometrical representation of the pure state space of a two-level quantum mechanical system}}
{{distinguish|Poincaré sphere (optics)}}
[[File:Bloch_sphere.svg|thumb|Bloch sphere]]

In quantum [[quantum mechanics|mechanics]] and [[Quantum computing|computing]], the '''Bloch sphere''' is a geometrical representation of the [[pure state]] space of a [[two-level system|two-level quantum mechanical system]] ([[qubit]]), named after the physicist [[Felix Bloch]].{{sfn | Bloch | 1946}}

Mathematically each quantum mechanical system is associated with a [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]] <math>H</math>. A pure state of a quantum system is represented by a non-zero vector <math>\psi </math> in <math>H</math>. As the vectors <math>\psi </math> and <math>\lambda \psi </math> (with <math>\lambda \in \mathbb{C}^*</math>) represent the same state, the level of the quantum system corresponds to the dimension of the Hilbert space and pure states can be represented as [[equivalence class]]es, or, '''[[Ray (quantum theory)|rays]]''' in a [[projective Hilbert space]] <math>\mathbf{P}(H_{n})=\mathbb{C}\mathbf{P}^{n-1}</math>.{{sfn|Bäuerle|de Kerf|1990|pp=330,341}} For a two-dimensional Hilbert space, the space of all such states is the [[complex projective line]] <math>\mathbb{C}\mathbf{P}^1.</math> This is the Bloch sphere, which can be mapped to the [[Riemann sphere]].

The Bloch sphere is a unit [[N-sphere|2-sphere]], with [[antipodal points]] corresponding to a pair of mutually orthogonal state vectors.  The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors <math>|0\rangle</math> and <math>|1\rangle</math>, respectively, which in turn might correspond e.g. to the [[Spin (physics)|spin]]-up and [[Spin (physics)|spin]]-down states of an electron.  This choice is arbitrary, however.  The points on the surface of the sphere correspond to the [[Quantum state#Superposition of pure states|pure state]]s of the system, whereas the interior points correspond to the [[Quantum state#Mixed states|mixed states]].{{sfn | Nielsen | Chuang | 2000}}<ref>{{Cite web|url=http://www.quantiki.org/wiki/Bloch_sphere|title = Bloch sphere &#124; Quantiki}}</ref>  The Bloch sphere may be generalized to an ''n''-level quantum system, but then the visualization is less useful.
 
The natural [[Metric (mathematics)|metric]] on the Bloch sphere is the [[Fubini–Study metric]].  The mapping from the unit 3-sphere in the two-dimensional state space <math>\mathbb{C}^2</math> to the Bloch sphere is the [[Hopf fibration]], with each [[Projective Hilbert space|ray]] of [[spinor]]s mapping to one point on the Bloch sphere.