dorsal/arxiv
View SchemaOne-and-a-half quantum de Finetti theorems
| Authors | Matthias Christandl, Robert Koenig, Graeme Mitchison, Renato Renner |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0602130 |
| URL | https://arxiv.org/abs/quant-ph/0602130 |
| DOI | 10.1007/s00220-007-0189-3 |
| Journal | Comm. Math. Phys., 273 (2), 473-498, (2007) |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
We prove a new kind of quantum de Finetti theorem for representations of the unitary group U(d). Consider a pure state that lies in the irreducible representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the symmetric representation, this yields the conventional quantum de Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second de Finetti-style theorem (our 'half' theorem); the de Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also provides some useful examples, and gives some insight into the structure of the set of convex combinations of product states.
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"abstract": "We prove a new kind of quantum de Finetti theorem for representations of the\nunitary group U(d). Consider a pure state that lies in the irreducible\nrepresentation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained\nin the tensor product of U_mu and U_nu; let xi be the state obtained by tracing\nout U_nu. We show that xi is close to a convex combination of states Uv, where\nU is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the\nsymmetric representation, this yields the conventional quantum de Finetti\ntheorem for symmetric states, and our method of proof gives near-optimal bounds\nfor the approximation of xi by a convex combination of product states. For the\nclass of symmetric Werner states, we give a second de Finetti-style theorem\n(our \u0027half\u0027 theorem); the de Finetti-approximation in this case takes a\nparticularly simple form, involving only product states with a fixed spectrum.\nOur proof uses purely group theoretic methods, and makes a link with the\nshifted Schur functions. It also provides some useful examples, and gives some\ninsight into the structure of the set of convex combinations of product states.",
"arxiv_id": "quant-ph/0602130",
"authors": [
"Matthias Christandl",
"Robert Koenig",
"Graeme Mitchison",
"Renato Renner"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s00220-007-0189-3",
"journal_ref": "Comm. Math. Phys., 273 (2), 473-498, (2007)",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "One-and-a-half quantum de Finetti theorems",
"url": "https://arxiv.org/abs/quant-ph/0602130"
},
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