Since Pauli matrices are traceless, it still holds that tr(ρ) = 1. However, by virtue of ( a ⋅ ⋅ σ σ ) ( b ⋅ ⋅ σ σ ) = ( a ⋅ ⋅ b ) I + i ( a × × b ) ⋅ ⋅ σ σ , ρ ρ 2 = 1 2 [ 1 2 ( 1 + | a | 2 ) I + a ⋅ ⋅ σ σ ] , hence tr ⁡ ⁡ ( ρ ρ 2 ) = 1 2 ( 1 + | a | 2 ) , which agrees with the fact that only states on the surface of the sphere itself are pure (i.e. | a | = 1 ).