Entanglement properties of positive operators with ranges in completely entangled subspaces

We prove that the projection on a completely entangled subspace $\mathcal{S}$ of maximum dimension obtained by Parthasarathy [K. R. Parthasarathy, Proc. Indian Acad. Sci. Math. Sci. 114, 365 (2004)] in a multipartite quantum system is not positive under partial transpose. We next show that a large number of positive operators with a range in $\mathcal{S}$ also have the same property. In this process we construct an orthonormal basis for $\mathcal{S}$ and provide a theorem to link the constructions of completely entangled subspaces due to Parthasarathy (as cited above), Bhat [B. V. R. Bhat, Int. J. Quantum Inf. 4, 325 (2006)], and Johnston [N. Johnston, Phys. Rev. A 87, 064302 (2013)].