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

Abstract

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ ${\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k$ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _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

$$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$

where ε is the collection of all completely entangled subspaces.

When${\mathcal{H}} = {\mathcal{H}}_2$ andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of${\mathcal{H}}_1 \otimes {\mathcal{H}}_2$ is given.

We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.