On the maximal dimension of a completely entangled subspace for finite level quantum systems

Let $\mathcalH_i$ be a finite dimensional complex Hilbert space of dimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i, 1,2, ..., k$. A subspace $S ⊂ \mathcalH = \mathcalH_A_1 A_2... A_k = \mathcalH_1 ⊗ \mathcalH_2 ⊗ ... ⊗ \mathcalH_k $ is said to be completely entangled if it has no nonzero product vector of the form $u_1 ⊗ u_2 ⊗ ... ⊗ u_k$ with $u_i$ in $\mathcalH_i$ for each $i$. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that $$\max_S ∈ \mathcalE \dim S = d_1 d_2... d_k - (d_1 + ... + d_k) + k - 1$$ where $\mathcalE $ is the collection of all completely entangled subspaces. When $\mathcalH_1 = \mathcalH_2 $ and $k = 2$ an explicit orthonormal basis of a maximal completely entangled subspace of $\mathcalH_1 ⊗ \mathcalH_2$ is given. We also introduce a more delicate notion of a perfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.