Apply the assumed <data-processing inequality for quantum relative entropy> to the normalized <partial trace> over $C$, with the two input states
$$
\rho_{ABC},
\qquad
\frac{I_A}{d_A}\otimes\rho_{BC}.
$$
The channel sends them to $\rho_{AB}\otimes I_C/d_C$ and $I_A/d_A\otimes\rho_B\otimes I_C/d_C$. Additivity over the common maximally mixed factor reduces data processing to
$$
D\left(\rho_{ABC}\middle\|\frac{I_A}{d_A}\otimes\rho_{BC}\right)
\geq
D\left(\rho_{AB}\middle\|\frac{I_A}{d_A}\otimes\rho_B\right).
$$
Expanding the <Umegaki relative entropy> in terms of <Von Neumann entropy> gives
$$
-S(\rho_{ABC})+\log d_A+S(\rho_{BC})
\geq
-S(\rho_{AB})+\log d_A+S(\rho_B).
$$
After cancelling $\log d_A$, this is precisely the 
$$
\boxed{S(\rho_B)+S(\rho_{ABC})
\leq S(\rho_{AB})+S(\rho_{BC})}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 3 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-323/3/i/solution