Pick a basis with states | 0 ⟩ ⟩ , | 1 ⟩ ⟩ in a two-dimensional Hilbert space, then the density operator is represented by the matrix ( ρ ρ i j ) = ( ρ ρ 00 ρ ρ 01 ρ ρ 10 ρ ρ 11 ) = ( p 0 ρ ρ 01 ρ ρ 01 ∗ ∗ p 1 )