Entanglement properties of positive operators with ranges in completely entangled subspaces

TL;DR

This paper investigates the entanglement properties of positive operators with ranges in maximally entangled subspaces, demonstrating their non-positivity under partial transpose and constructing bases linking various entangled subspace constructions.

Contribution

It proves the non-positivity under partial transpose of projections on maximally entangled subspaces and constructs an orthonormal basis linking different entangled subspace methods.

Findings

Projection on maximally entangled subspace is not positive under partial transpose

Several positive operators with range in such subspaces also lack positivity under partial transpose

Constructs an orthonormal basis linking different entangled subspace constructions

Abstract

We prove that the projection on a completely entangled subspace S of maximum dimension in a multipartite quantum system obtained by Parthasarathy is not positive under partial transpose. We next show that several positive operators with range in S also have the same property. In this process we construct an orthonormal basis of S and provide a linking theorem to link the constructions of completely entangled subspaces due to Parthasarthy, Bhat and Johnston.