Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 323 / 5 / 1 / Solution

Use logarithms to base two, so Von Neumann entropy and quantum relative entropy are measured in bits. For density operators $\rho ,\sigma$,The quantum relative entropy formula applies when the support of a positive operator $\rho$ is contained in that of $\sigma$; otherwise $D(\rho \Vert \sigma )=+\infty$. Use $0{\log }_{2}0=0$ in the Von Neumann entropy formula.

The binary density operator $\omega [p]$ is diagonal with eigenvalues $1-p,p$. Its Von Neumann entropy is the binary entropy:The two diagonal density operators commute, so their quantum relative entropy isA term with zero numerator has value zero; a positive numerator and zero denominator give $+\infty$. In particular, $D(\omega [0]\Vert \omega [0])=D(\omega [1]\Vert \omega [1])=0$, whereas $q=0,p>0$ or $q=1,p<1$ gives $+\infty$.