Density Operators and Ensembles

• Notation. What will here be called density operators are often (as in CQT) referred to as density matrices. Alas, a “density matrix” is not always a matrix, and has nothing whatsoever to do with “density” in the usual meaning of that term. Thus the term “density operator,” which only suffers from the second of these defects, is to be preferred. ⋆ Mathematical definition: any operator ρ on the Hilbert space H which is positive (semidefinite) and has trace 1 is a density operator. ◦ Positive: ρ = ρ is Hermitian, and none of its eigenvalues is negative. • If one eigenvalue is 1 and the rest are zero, ρ = |ψ〉〈ψ| for some normalized |ψ〉, and is called a pure state. Otherwise it represents a mixed state.