If ρ ρ is a projection, which defines a pure state, then the upper bound is saturated: tr ⁡ ⁡ ( ρ ρ 2 ) = tr ⁡ ⁡ ( ρ ρ ) = 1 (see Projections). The lower bound is obtained by the completely mixed state, represented by the matrix 1 d I d .