= 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. \b[The state is therefore entangled throughout the specified open interval.]
Equivalently, the <reduced density matrix> is
$$
\rho_A=\begin{pmatrix}1-p&0\\0&p\end{pmatrix},\qquad
\operatorname{Tr}(\rho_A^2)=1-2p(1-p)<1.
$$
The 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 $|00\rangle$ or $|11\rangle$.
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