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Genuine high-dimensional entanglement, i.e. the property of having a highSchmidt number, constitutes a resource in quantum communication, overcominglimitations of low-dimensional systems. States with a positive partialtranspose (PPT), on the other hand, are generally considered weakly entangled,as they can never be distilled into pure entangled states. This naturallyraises the question, whether high Schmidt numbers are possible for PPT states.Volume estimates suggest that optimal, i.e. linear, scaling in local dimensionshould be possible, albeit without providing an insight into the possibleslope. We provide the first explicit construction of a family of PPT statesthat achieves linear scaling in local dimension and we prove that random PPTstates typically share this feature. Our construction also allows us to answera recent question by Chen et al. on the existence of PPT states whose Schmidtnumber increases by an arbitrarily large amount upon partial transposition.Finally, we link the Schmidt number to entangled sub-block matrices of aquantum state. We use this connection to prove that quantum states invariantunder partial transposition on the smaller of their two subsystems cannot havemaximal Schmidt number. This generalizes a well-known result by Kraus et al. Wealso show that the Schmidt number of absolutely PPT states cannot be maximal,contributing to an open problem in entanglement theory.

High-Dimensional Entanglement in States with Positive Partial Transposition