dorsal/arxiv
View SchemaHilbert's 17th problem and the quantumness of states
| Authors | J. Korbicz, J. I. Cirac, J. Wehr, M. Lewenstein |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408029 |
| URL | https://arxiv.org/abs/quant-ph/0408029 |
| DOI | 10.1103/PhysRevLett.94.153601 |
| Journal | Phys. Rev. Lett. 94, 153601 (2005) |
Abstract
A state of a quantum systems can be regarded as {\it classical} ({\it quantum}) with respect to measurements of a set of canonical observables iff there exists (does not exist) a well defined, positive phase space distribution, the so called Galuber-Sudarshan $P$-representation. We derive a family of classicality criteria that require that averages of positive functions calculated using $P$-representation must be positive. For polynomial functions, these criteria are related to 17-th Hilbert's problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as {\it non-classicality witnesses}. We show that every generic non-classical state can be detected by a polynomial that is a sum of squares of other polynomials (sos). We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sos polynomial that detects them is introduced. Polynomial non-classicality witnesses can be directly measured.
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"abstract": "A state of a quantum systems can be regarded as {\\it classical} ({\\it\nquantum}) with respect to measurements of a set of canonical observables iff\nthere exists (does not exist) a well defined, positive phase space\ndistribution, the so called Galuber-Sudarshan $P$-representation. We derive a\nfamily of classicality criteria that require that averages of positive\nfunctions calculated using $P$-representation must be positive. For polynomial\nfunctions, these criteria are related to 17-th Hilbert\u0027s problem, and have\nphysical meaning of generalized squeezing conditions; alternatively, they may\nbe interpreted as {\\it non-classicality witnesses}. We show that every generic\nnon-classical state can be detected by a polynomial that is a sum of squares of\nother polynomials (sos). We introduce a very natural hierarchy of states\nregarding their degree of quantumness, which we relate to the minimal degree of\na sos polynomial that detects them is introduced. Polynomial non-classicality\nwitnesses can be directly measured.",
"arxiv_id": "quant-ph/0408029",
"authors": [
"J. Korbicz",
"J. I. Cirac",
"J. Wehr",
"M. Lewenstein"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevLett.94.153601",
"journal_ref": "Phys. Rev. Lett. 94, 153601 (2005)",
"title": "Hilbert\u0027s 17th problem and the quantumness of states",
"url": "https://arxiv.org/abs/quant-ph/0408029"
},
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