Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 33B / b / iv / Solution

The eigenvalues found in part (ii) give the purity of a density operatorThe displayed truncation lies between zero and one when $|\lambda |\le 2\sqrt{2}{\omega }^2$. Perturbation theory requires the stronger condition $|\lambda |\ll {\omega }^2$; the exact coupled oscillator is stable only for $|\lambda |<{\omega }^2$. The reduction in purity records entanglement between $A$ and $B$.