= Local-unitary equivalence
{title2=$|\Psi\rangle=(U\otimes V)|\Phi\rangle$}
= Local-unitary equivalent
{synonym}
= LU-equivalence
{c}
{synonym}
Two bipartite <pure states> on fixed subsystem spaces are <local-unitary equivalent> when related by the displayed product of <unitary operators>. <Identity operators> give <reflexivity>, inverse operators give symmetry, and products of local operators give <transitivity>. Thus this is an <equivalence relation>, whose classes are the orbits under local unitaries.
Every pair of normalized <product states> is <local-unitary equivalent>: extend each factor to an <orthonormal basis> and use a <unitary operator> carrying one <basis> to the other. Local operators preserve a product factorization, and their inverses do too, so <quantum entanglement> cannot be created or removed this way. More generally, the <tensor-product coefficient matrix> changes as $C\mapsto UCV^T$, which preserves its <singular values> and therefore its <Schmidt coefficients>. The transpose on the second factor is ordinary, not a <conjugate transpose>, because coefficients of a <tensor product> depend bilinearly on the two input <vectors>.
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