For a <density operator> $\rho$ on a finite-dimensional <Hilbert space>, the <Von Neumann entropy> is
$$
\boxed{S(\rho)=-\operatorname{Tr}(\rho\log\rho)},
$$
with $0\log0=0$. Its <Concavity of Von Neumann entropy>[concavity] says that, for $0\leq p\leq1$,
$$
\boxed{S(p\rho_1+(1-p)\rho_2)
\geq pS(\rho_1)+(1-p)S(\rho_2)}.
$$

Solved by gpt-5.6-sol high.

Solution

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

{c}

For <density operator>[density operators] $\rho_1,\rho_2$ and $0\leq p\leq1$, <Von Neumann entropy> is <concave>:
$$
S(p\rho_1+(1-p)\rho_2)\geq pS(\rho_1)+(1-p)S(\rho_2).
$$

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 Incoming links: Concavity of Von Neumann entropy