Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 2

An ensemble of normalized quantum states with probabilities $p_j$ is represented by the density operatorIt determines every observable expectation value through $⟨A⟩=\mathrm{Tr}(\rho A)$. The ensemble decomposition need not be unique; the operator specifies the measurement statistics.

Each term is a Hermitian rank-one projection and $p_j$ is real, so ${\rho }^{†}=\rho$. Using the rank-one trace,The remaining admissibility condition follows from $⟨v|\rho |v⟩={\sum }_j{p}_j|⟨{\psi }_j|v⟩{|}^2\ge 0$: $\rho$ is positive semidefinite.

A normalized pure state has $\rho =|\psi ⟩⟨\psi |$, and thereforeConversely, diagonalize a permissible density operator. Its eigenvalues obey $0\le {\lambda }_j\le 1$ and ${\sum }_j{\lambda }_j=1$, soEquality requires every eigenvalue to be zero or one; exactly one is one. Thus the idempotent matrix condition and purity one both characterize a pure state. A mixed state has at least two positive eigenvalues, ${\rho }^2\ne \rho$ and $\mathrm{Tr}{\rho }^2<1$. In dimension $d$ its purity is at least $1/d$.

The given real symmetric matrix is Hermitian and has trace one. A spectral decomposition supplies its positivity without relying on diagonal entries alone:The remaining normalized eigenvector is $(-1,1,1{)}^{\mathsf{T}}/\sqrt{3}$ with eigenvalue zero. Consequently the spectrum is $\{3/4,1/4,0\}$, every eigenvalue is nonnegative, and the matrix is permissible. Its purity is $(3/4{)}^2+(1/4{)}^2=\boxed{5/8}$, so it is mixed.

The original Von Neumann entropy depends on its eigenvalues, with $0\log 0$ defined by continuity:After rank-one dephasing, the eigenvalues are the retained diagonal populations $1/2,1/4,1/4$, givingdephasing removes coherences in the specified basis, increasing the system's mixedness while preserving those measurement probabilities. It cannot be a closed-system unitary transformation, since the eigenvalues change. The subsystem entropy increase does not by itself assert an increase in the entropy of a closed system plus its environment.