 Solution

The Schmidt decomposition of a normalized bipartite pure state isThe two sets of local vectors are orthonormal. The Schmidt coefficients are the nonnegative numbers $\sqrt{{\lambda }_j}$, and the Schmidt number for this pure state, also called its Schmidt rank, is $r$, the number of nonzero coefficients. It equals the rank of either reduced density operator and of the coefficient matrix in any product bases. The general mixed-state Schmidt number specializes to this same value on a pure state.