Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 48 / 5 / d / Solution

The two displayed amplitudes have modulus $1/\sqrt{2}$. Absorb their phases into Bob's basis rather than use complex Schmidt coefficients. Take $|u_0⟩=|0⟩$, $|u_1⟩=|1⟩$, andBoth local bases are orthonormal. A Schmidt decomposition is thereforeThe two Schmidt coefficients are $1/\sqrt{2}$, the Schmidt number is two, and both reduced density operators are $I/2$. Their equal eigenvalues make this decomposition nonunique.