dorsal/arxiv
View SchemaAlgebraic measures of entanglement
| Authors | Jean-Luc Brylinski |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0008031 |
| URL | https://arxiv.org/abs/quant-ph/0008031 |
Abstract
We study the rank of a general tensor $u$ in a tensor product $H_1\ot...\ot H_k$. The rank of $u$ is the minimal number $p$ of pure states $v_1,...,v_p$ such that $u$ is a linear combination of the $v_j$'s. This rank is an algebraic measure of the degree of entanglement of $u$. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in $(\C^2)^{\ot 3}$ and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for $(\C^2)^{\ot 4}$; in particular, we show that the maximal rank of a tensor in $(\C^2)^{\ot 4}$ is equal to 4.
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"abstract": "We study the rank of a general tensor $u$ in a tensor product $H_1\\ot...\\ot\nH_k$. The rank of $u$ is the minimal number $p$ of pure states $v_1,...,v_p$\nsuch that $u$ is a linear combination of the $v_j$\u0027s. This rank is an algebraic\nmeasure of the degree of entanglement of $u$. Motivated by quantum computation,\nwe completely describe the rank of an arbitrary tensor in $(\\C^2)^{\\ot 3}$ and\ngive normal forms for tensor states up to local unitary transformations. We\nalso obtain partial results for $(\\C^2)^{\\ot 4}$; in particular, we show that\nthe maximal rank of a tensor in $(\\C^2)^{\\ot 4}$ is equal to 4.",
"arxiv_id": "quant-ph/0008031",
"authors": [
"Jean-Luc Brylinski"
],
"categories": [
"quant-ph"
],
"title": "Algebraic measures of entanglement",
"url": "https://arxiv.org/abs/quant-ph/0008031"
},
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"type": "Model",
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