Is absolute separability determined by the partial transpose?

The absolute separability problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU† is separable for all unitary matrices U. We provide evidence that ρ is absolutely separable if and only if UρU† has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.