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LetH ibe a finite dimensional complex Hilbert space of dimensiond i associated with a finite level quantum system Ai 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 ui 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.

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