 Solution

In the computational product bases, the coefficient matrix isIts determinant is $-1/3$, so it has rank two. Therefore the Schmidt number is $\boxed{2}$. As a check, the reduced density operator $C{C}^{†}=\frac{1}{3}\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3+\sqrt{5})/6$ and $(3-\sqrt{5})/6$, both positive.