dorsal/arxiv
View SchemaOn the maximal dimension of a completely entangled subspace for finite level quantum systems
| Authors | K. R. Parthasarathy |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0405077 |
| URL | https://arxiv.org/abs/quant-ph/0405077 |
Abstract
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.
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"abstract": "Let $\\mathcal{H}_i$ be a finite dimensional complex Hilbert space of\ndimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i,\n1,2, ..., k$. A subspace $S \\subset \\mathcal{H} = \\mathcal{H}_{A_{1}\n A_{2}... A_{k}} = \\mathcal{H}_1 \\otimes \\mathcal{H}_2 \\otimes ... \\otimes\n\\mathcal{H}_k $ is said to be {\\it completely entangled} if it has no nonzero\nproduct vector of the form $u_1 \\otimes u_2 \\otimes ... \\otimes u_k$ with $u_i$\nin $\\mathcal{H}_i$ for each $i$. Using the methods of elementary linear algebra\nand the intersection theorem for projective varieties in basic algebraic\ngeometry we prove that $$\\max_{S \\in \\mathcal{E}} \\dim S = d_1 d_2... d_k -\n(d_1 + ... + d_k) + k - 1$$ where $\\mathcal{E} $ is the collection of all\ncompletely entangled subspaces.\n When $\\mathcal{H}_1 = \\mathcal{H}_2 $ and $k = 2$ an explicit orthonormal\nbasis of a maximal completely entangled subspace of $\\mathcal{H}_1 \\otimes\n\\mathcal{H}_2$ is given.\n We also introduce a more delicate notion of a {\\it perfectly entangled}\nsubspace for a multipartite quantum system, construct an example using the\ntheory of stabilizer quantum codes and pose a problem.",
"arxiv_id": "quant-ph/0405077",
"authors": [
"K. R. Parthasarathy"
],
"categories": [
"quant-ph"
],
"title": "On the maximal dimension of a completely entangled subspace for finite level quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0405077"
},
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