Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. 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 one-dimensional subspaces of the corresponding Hilbert space (or the 'points' of the projective Hilbert space).For a two-dimensional Hilbert space, this is simply the complex projective line ℂℙ1. This is the Bloch sphere. The Bloch sphere is a unit 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 | 0 ⟩ {displaystyle |0 angle } and | 1 ⟩ {displaystyle |1 angle } , respectively,which in turn might correspond e.g. to the spin-up and spin-down states of an electron.This choice is arbitrary, however.The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states.The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful. For historical reasons, in optics the Bloch 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 ℂ2 to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere. Given an orthonormal basis, any pure state | ψ ⟩ {displaystyle |psi angle } of a two-level quantum system can be written as a superposition of the basis vectors | 0 ⟩ {displaystyle |0 angle } and | 1 ⟩ {displaystyle |1 angle } , where the coefficient or amount of each basis vector is a complex number.Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of | 0 ⟩ {displaystyle |0 angle } to be real and non-negative. We also know from quantum mechanics that the total probability of the system has to be one: ⟨ ψ | ψ ⟩ = 1 {displaystyle langle psi |psi angle =1} , or equivalently | | | ψ ⟩ | | 2 = 1 {displaystyle ||,|psi angle ,||^{2}=1} . Given this constraint, we can write | ψ ⟩ {displaystyle |psi angle } using the following representation: Except in the case where | ψ ⟩ {displaystyle |psi angle } is one of the ket vectors (see Bra-ket notation) | 0 ⟩ {displaystyle |0 angle } or | 1 ⟩ {displaystyle |1 angle } ,the representation is unique. The parameters θ {displaystyle heta ,} and ϕ {displaystyle phi ,} , re-interpreted in spherical coordinates as respectively the colatitude with respect to the z-axis and the longitude with respect to the x-axis, specify a point on the unit sphere in R 3 {displaystyle mathbb {R} ^{3}} .