Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 48 / 2 / c / Solution

After the first quantum channel, denote the joint reference-output state by ${\omega }_{R{B}_1}$. Dilate the second CPTP map as an linear isometry $B_1\to B_2{E}_2$, with the resulting state denoted $\tau$. Linear isometries preserve nonzero eigenvalues, both globally and on the transformed output, givingTracing out $E_2$ implements the second channel, soThe Strong subadditivity of Von Neumann entropy for $R,B_2,E_2$ gives $S(R{B}_2)+S(B_2{E}_2)\ge S(B_2)+S(R{B}_2{E}_2)$. Rearranging proves the data-processing inequality for coherent information: