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Proceedings Mathematical Sciences

, Volume 114, Issue 4, pp 365–374 | Cite as

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

  • K. R. ParthasarathyEmail author
Regular Articles

Abstract

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 1u 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.

Keywords

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

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References

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Copyright information

© Indian Academy of Sciences 2004

Authors and Affiliations

  1. 1.Indian Statistical InstituteDelhi CentreNew DelhiIndia

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