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.

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