Bloch Sphere - Definition

Definition

Given an orthonormal basis, any pure state of a two-level quantum system can be written as a superposition of the basis vectors and, 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 to be real and non-negative. We also know from quantum mechanics that the total probability of the system has to be one, so it must be that . Given this constraint, we can write in the following representation:

|\psi\rangle = \cos\left(\tfrac{\theta}{2}\right) |0 \rangle \, + \, e^{i \phi} \sin\left(\tfrac{\theta}{2}\right) |1 \rangle = \cos\left(\tfrac{\theta}{2}\right) |0 \rangle \, + \, ( \cos \phi + i \sin \phi) \, \sin\left(\tfrac{\theta}{2}\right) |1 \rangle

with and .

Except in the case where is one of the ket vectors or the representation is unique. The parameters and, re-interpreted as spherical coordinates, specify a point on the unit sphere in . For mixed states, any two-dimensional density operator can be expanded using the identity and the Hermitian, traceless Pauli matrices :

,

where is called the Bloch vector of the system. The eigenvalues of are given by . As density operators must be positive-semidefinite, we have .

For pure states we must have

,

in accordance with the previous result. Hence the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

Read more about this topic:  Bloch Sphere

Famous quotes containing the word definition:

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)