dorsal/arxiv
View SchemaCatalytic majorization and $\ell_p$ norms
| Authors | Guillaume Aubrun, Ion Nechita |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0702153 |
| URL | https://arxiv.org/abs/quant-ph/0702153 |
| DOI | 10.1007/s00220-007-0382-4 |
| Journal | Communications in Mathematical Physics 278 (2008) 133-144 |
Abstract
An important problem in quantum information theory is the mathematical characterization of the phenomenon of quantum catalysis: when can the surrounding entanglement be used to perform transformations of a jointly held quantum state under LOCC (local operations and classical communication) ? Mathematically, the question amounts to describe, for a fixed vector $y$, the set $T(y)$ of vectors $x$ such that we have $x \otimes z \prec y \otimes z$ for some $z$, where $\prec$ denotes the standard majorization relation. Our main result is that the closure of $T(y)$ in the $\ell_1$ norm can be fully described by inequalities on the $\ell_p$ norms: $\|x\|_p \leq \|y\|_p$ for all $p \geq 1$. This is a first step towards a complete description of $T(y)$ itself. It can also be seen as a $\ell_p$-norm analogue of Ky Fan dominance theorem about unitarily invariant norms. The proofs exploits links with another quantum phenomenon: the possibiliy of multiple-copy transformations ($x^{\otimes n} \prec y^{\otimes n}$ for given $n$). The main new tool is a variant of Cram\'er$ theorem on large deviations for sums of i.i.d. random variables.
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"abstract": "An important problem in quantum information theory is the mathematical\ncharacterization of the phenomenon of quantum catalysis: when can the\nsurrounding entanglement be used to perform transformations of a jointly held\nquantum state under LOCC (local operations and classical communication) ?\nMathematically, the question amounts to describe, for a fixed vector $y$, the\nset $T(y)$ of vectors $x$ such that we have $x \\otimes z \\prec y \\otimes z$ for\nsome $z$, where $\\prec$ denotes the standard majorization relation. Our main\nresult is that the closure of $T(y)$ in the $\\ell_1$ norm can be fully\ndescribed by inequalities on the $\\ell_p$ norms: $\\|x\\|_p \\leq \\|y\\|_p$ for all\n$p \\geq 1$. This is a first step towards a complete description of $T(y)$\nitself. It can also be seen as a $\\ell_p$-norm analogue of Ky Fan dominance\ntheorem about unitarily invariant norms. The proofs exploits links with another\nquantum phenomenon: the possibiliy of multiple-copy transformations\n($x^{\\otimes n} \\prec y^{\\otimes n}$ for given $n$). The main new tool is a\nvariant of Cram\\\u0027er$ theorem on large deviations for sums of i.i.d. random\nvariables.",
"arxiv_id": "quant-ph/0702153",
"authors": [
"Guillaume Aubrun",
"Ion Nechita"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s00220-007-0382-4",
"journal_ref": "Communications in Mathematical Physics 278 (2008) 133-144",
"title": "Catalytic majorization and $\\ell_p$ norms",
"url": "https://arxiv.org/abs/quant-ph/0702153"
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