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

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## Abstract

Let where ε is the collection of all completely entangled subspaces.

*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 ⊂*\({\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k \) is said to be*completely entangled*if it has no non-zero product vector of the form*u*_{1}⊗*u*_{2}⊗ ... ⊗*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$$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$

When\({\mathcal{H}} = {\mathcal{H}}_2 \) 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*perfectly entangled* subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

## Keywords

Finite level quantum systems separable states entangled states completely entangled subspaces perfectly entangled subspace stabilizer quantum code## Preview

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## References

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© Indian Academy of Sciences 2004