Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 4 / e

The normalized singlet state has column vector $(0,1,-1,0{)}^{\mathsf{T}}/\sqrt{2}$ in the specified product basis. Its outer product isIt obeys ${\rho }_D^2={\rho }_D$ and has eigenvalues $(1,0,0,0)$, hence purity of a density operator one. The joint two-atom system is therefore a pure state, even though the quantum entanglement prevents assigning a separate pure spin state to each atom.

Taking the partial trace over the left atom in the singlet state gives the right-hand reduced density matrixThus every measurement on the right stream alone has the same statistics as B or C. The joint pure state is compatible with locally mixed states; the correlations reside in the full density operator.

The reduced density matrix has two eigenvalues equal to $1/2$. Its Von Neumann entropy, which is the entanglement entropy of the bipartite pure state, isThis is one bit when base-two logarithms are used. It is the maximum entanglement entropy for a pair of two-dimensional systems, while the joint Von Neumann entropy of the singlet state remains zero.