Let $\mathcal{ H}_i$ 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 \subset \mathcal{ H} = \mathcal{ H}_{ A_{ 1} A_{ 2}... A_{ k}} = \mathcal{ H}_1 \otimes \mathcal{ H}_2 \otimes ... \otimes \mathcal{ H}_k $ is said to be { \it completely entangled} if it has no nonzero product vector of the form $u_1 \otimes u_2 \otimes ... \otimes u_k$ with $u_i$ in $\mathcal{ H}_i$ for each $i$. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that $$\max_{ S \in \mathcal{ E}} \dim S = d_1 d_2... d_k - (d_1 + ... + d_k) + k - 1$$ where $\mathcal{ E} $ is the collection of all completely entangled subspaces. When $\mathcal{ H}_1 = \mathcal{ H}_2 $ and $k = 2$ 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 a { \it perfectly entangled} subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.
Categories: quant-ph