6.1 Definitions

If you are an impatient mathematically minded person, who feels more comfortable when things are properly defined right from the beginning, here is your definition:85

A density operator \rho on a finite dimensional Hilbert space \mathcal{H} is any non-negative self-adjoint operator with trace equal to one.

It follows that any such \rho can always be diagonalised, that the eigenvalues are all real and non-negative, and that the eigenvalues sum to one. Moreover, given two density operators \rho_1 and \rho_2, we can always construct another density operator as a convex sum of the two: \rho = p_1\rho_1 + p_2\rho_2 \qquad\text{where}\quad p_1, p_2 \geqslant 0 \text{ and } p_1+p_2 = 1. You should check that \rho has all the defining properties of a density matrix, i.e. that it is self-adjoint, non-negative, and that its trace is one. This means that density operators form a convex set.86

An important example of a density operator is a rank one projector.87 Any quantum state that can be described by the state vector |\psi\rangle, called a pure state, can be also described by the density operator \rho=|\psi\rangle\langle\psi|. Pure states are the extremal points in the convex set of density operators: they cannot be expressed as a convex sum of other elements in the set. In contrast, all other states, called mixed states, can be always written as the convex sum of pure states: \sum_i p_i |\psi_i\rangle\langle\psi_i| (p_i\geqslant 0 and \sum_i p_i=1). Now that we have cleared the mathematical essentials, we will turn to physical applications.

  1. A self-adjoint matrix M is said to be non-negative, or positive semi-definite, if \langle v|M|v\rangle\geqslant 0 for any vector |v\rangle, or if all of its eigenvalues are non-negative, or if there exists a matrix A such that M=A^\dagger A. (This is called a Cholesky factorization.)↩︎

  2. A subset of a vector space is said to be convex if, for any two points in the subset, the straight line segment joining them is also entirely contained inside the subset.↩︎

  3. The rank of a matrix is the number of its non-zero eigenvalues.↩︎