### Abstract

A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.

Original language | English (US) |
---|---|

Article number | 153601 |

Journal | Physical Review Letters |

Volume | 94 |

Issue number | 15 |

DOIs | |

State | Published - Apr 22 2005 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physical Review Letters*,

*94*(15), [153601]. https://doi.org/10.1103/PhysRevLett.94.153601

**Hilbert's 17th problem and the quantumness of states.** / Korbicz, J. K.; Cirac, J. I.; Wehr, Jan; Lewenstein, M.

Research output: Contribution to journal › Article

*Physical Review Letters*, vol. 94, no. 15, 153601. https://doi.org/10.1103/PhysRevLett.94.153601

}

TY - JOUR

T1 - Hilbert's 17th problem and the quantumness of states

AU - Korbicz, J. K.

AU - Cirac, J. I.

AU - Wehr, Jan

AU - Lewenstein, M.

PY - 2005/4/22

Y1 - 2005/4/22

N2 - A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.

AB - A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.

UR - http://www.scopus.com/inward/record.url?scp=18144365184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18144365184&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.94.153601

DO - 10.1103/PhysRevLett.94.153601

M3 - Article

AN - SCOPUS:18144365184

VL - 94

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 15

M1 - 153601

ER -