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:

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)