Extensions of Completely Entangled Subspaces in C2⊗CN

In the Kronecker product C2⊗CN, completely entangled subspaces (CESs) are subspaces containing no nonzero product vectors. In this paper, we study the extension of a given M-dimensional CES to a higher-dimensional CES. We first prove an existence result using the geometry of the Segre variety and an incidence-variety dimension-counting argument. In particular, an M-dimensional CES can be extended to an (M+1)-dimensional CES whenever M<N−1. We then introduce a constructive method based on matrix representations and invertible local transformations. We show that if the corresponding matrix Ψ satisfies rank(Ψ)≤N−2, an explicit extension can be constructed. In particular, the condition 2M≤N−2 is sufficient for this construction. Finally, we give a concrete example in C2⊗C8, where a three-dimensional CES is extended to a four-dimensional CES.