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:
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:
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)