Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / b

No: the advertised pure joint state is incompatible with these marginals. Their eigenvalue multisets are $\{0.9,0.1\}$ and $\{0.7,0.3\}$, which differ. Part a shows that a bipartite pure state must give the same nonzero Schmidt spectrum on both sides.

This excludes Eve's purity claim, not every possible joint state or every possibility of entanglement. A mixed state can have unequal marginal spectra; the product ${\rho }_A\otimes {\rho }_B$, for example, has exactly the stated marginals.