A density matrix is a positive semidefinite Hermitian matrix of trace 1. It represents a quantum state on a finite-dimensional Hilbert space. Its eigenvalues are nonnegative and sum to 1, so they can be interpreted as probabilities in a decomposition into orthogonal pure states.
A pure state has a rank-one density matrix 
for a unit vector 
, and satisfies 
. A mixed state is not pure. Joint density matrices
on a tensor product can be separable or entangled.
