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

Abstract

Let H_i be a finite dimensional complex Hilbert space of dimension d_i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspace S ⊂ H = H_A1A2......_AK = H₁ ⊗ H₂ ⊗......⊗H_kis said to be completely entangled if it has no non-zero product vector of the form u₁⊗ u₂ ⊗ ... ⊗ u_k with u_i in H_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∈εdim S= d₁d₂.......d_k-(d₁+...+d_k)+k-1 where ε is the collection of all completely entangled subspaces. When H₁=H₂ and k = 2 an explicit orthonormal basis of a maximal completely entangled subspace of H₁ ⊗ H₂is 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.