= Local-unitary invariance of reduced-state spectra
{title2=$\rho_A\mapsto U_A\rho_AU_A^\dagger$}
Under a product <unitary operator> $U_A\otimes U_B$, the <reduced density matrix> of $A$ becomes $U_A\rho_AU_A^\dagger$. The unitary on $B$ cancels under the <partial trace>. Unitary conjugation preserves <eigenvalues>, so the squared <Schmidt coefficients> of a bipartite <pure state> are unchanged. Conversely, two normalized bipartite pure states with the same squared Schmidt coefficients, including multiplicities and zero padding, have <Schmidt decompositions> carried into each other by local unitaries mapping their Schmidt bases. Thus reduced-state spectra completely classify bipartite pure states up to local unitary operations.
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