Theoretical Statistics and Mathematics Unit, ISI Delhi

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

by K. R. Parthasarathy

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 $i = i, 1,2, \ldots , k$. A subspace $S \subset \mathcal{H} = \mathcal{H} _{A_{1} A_{2}\ldots A_{k}} = \mathcal{H} _1 \otimes \mathcal{H} _2 \otimes \ldots \otimes \mathcal{H} _k $ is said to be {\it completely entangled} if it has no nonzero product vector of the form $u_1 \otimes u_2 \otimes \ldots \otimes u_k$ 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 $$\max _{S \in \mathcal{E} } \dim S = d_1 d_2\ldots d_k - (d_1 + \cdots +
d_k) + k - 1$$
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.

isid/ms/2004/06 [fulltext]

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