Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 2 / a / ii / Solution

The displayed computational basis expansion is already a Schmidt decomposition, with coefficients $\sqrt{1-p}$ and $\sqrt{p}$. For $0<p<1$ both are nonzero, so the Schmidt rank is two. A pure product state has Schmidt rank one. The state is therefore entangled throughout the specified open interval.

Equivalently, the reduced density matrix isThe reduced state is a mixed state, which is impossible for one factor of a pure product state. At the excluded endpoints the Schmidt rank drops to one, giving $|00⟩$ or $|11⟩$.