Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 4 / b

Put $\rho =p{\rho }_1+(1-p){\rho }_2$. The operator inequality $\rho \ge p{\rho }_1$ and the operator monotonicity of logarithm give, on the support of ${\rho }_1$,Consequently,The corresponding inequality from $\rho \ge (1-p){\rho }_2$ yieldsAdding these inequalities and using $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$ proves the entropy bound for a binary mixture:Singular states follow by adding a positive multiple of the identity and taking a limit; the endpoint cases use $0\log 0=0$.

Solved by gpt-5.6-sol high.