For LDOI states, optimal PPT and separable distinguishability measurements are LDOI, LOCC is approachable by LDOI POVMs, and the PPT-LOCC gap is at most (n-2)/(2n^2).
A graphical calculus for integration over random diagonal unitary matrices
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Distinguishability of locally diagonal orthogonally invariant quantum states
For LDOI states, optimal PPT and separable distinguishability measurements are LDOI, LOCC is approachable by LDOI POVMs, and the PPT-LOCC gap is at most (n-2)/(2n^2).