On the Maximal Dimension of a Completely Entangled Subspace for Finite Level Quantum Systems

[摘要] 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 $i=1, 2,ldots,k$. A subspace $Ssubsetmathcal{H} = mathcal{H}_{A_1 A_2ldots A_k} = mathcal{H}_1 otimes mathcal{H}_2 otimescdotsotimes mathcal{H}_k$ is said to be completely entangled if it has no non-zero product vector of the form $u_1 otimes u_2 otimescdotsotimes u_k$ 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$$maxlimits_{Sinmathcal{E}}dim S=d_1 d_2ldots d_k-(d_1+cdots +d_k)+k-1,$$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 perfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

[学科分类] 数学（综合）

[关键词] Finite level quantum systems;separable states;entangled states;completely entangled subspaces;perfectly entangled subspace;stabilizer quantum code.