Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 51 / 3 / a / Solution

Put $s={\sin }^2\alpha$ and $t={\sin }^2\beta$. Then $0<s<t<1/2$, and the ordered squared Schmidt coefficients of the source and target are $(1-s,s)$ and $(1-t,t)$.

Nielsen's pure-state conversion theorem states that a deterministic exact conversion between bipartite pure states by LOCC exists precisely when the source's vector of squared Schmidt coefficients is majorized by the target's vector, with zero padding if needed. In the convention $x\prec y$, majorization requires ${\sum }_{j=1}^k{x}_j^{\downarrow }\le {\sum }_{j=1}^k{y}_j^{\downarrow }$ for every $k$, and equality of total sums. Here the first inequality would require $1-s\le 1-t$, but $s<t$ gives the reverse strict inequality. Thus exact conversion with certainty is impossible.