Can two states with the same entanglement be transformed into each other using local unitaries?

Question

Take two pure bi-partite states $\psi$ and $\phi$ that have the same amount of entanglement in them as quantified by concurrence (does the measure make a difference?). Can any such states be transformed into each other using local unitaries?

Answer 1

Any two bipartite pure states $\psi$ and $\phi$ can be transformed into each other with local unitaries if and only if they have the same Schmidt coefficients. (To prove the 'only if' part, note that the reduced density matrix of either qubit has eigenvalues that are the squares of the Schmidt coefficients, and are unchanged by unitaries).

The concurrence is typically specified for two-qubit states. Any reasonable entanglement monotone on two qubits, when calculated for pure states, must be a decreasing function of the largest Schmidt coefficient of the state, and thus two states having the same entanglement would have the same Schmidt coefficients, and hence be unitarily equivalent.

If you generalise beyond qubits, there will be entanglement measures that do not contain as much fine-grained information as the Schmidt coefficients, and so equality of that entanglement measure will be necessary but not sufficient for unitary equivalence.