Let ρ be a positive real number. In order that there exist complex numbers ψ0 and ψ1 such that $$ |\psi _0 + \psi _1 |^2 = 1 $$ (1) $$ \psi _0 \psi _0^* + \psi _0^* \psi _1 = 1 - \rho $$ (2) it is necessary and sufficient that 2ρ≥ 1. Moreover if 2ρ≥ 1 and ρ0, ρ1 are non-negative quantities such that ρ0+ρ1=ρ then there are solutions ψ0, ψ1 to (1) and (2) with |ψ0|2 = ρ0, |ψ1|2=ρ1 if and only if $$ \left| {\rho _0 - \rho _1 } \right| \leqslant \left( {2\rho - 1} \right)^{1/2} $$ (3)