Hilbert's 17th problem and the quantumness of states

J. K. Korbicz, J. I. Cirac, Jan Wehr, M. Lewenstein

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37 Scopus citations


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 languageEnglish (US)
Article number153601
JournalPhysical review letters
Issue number15
StatePublished - Apr 22 2005

ASJC Scopus subject areas

  • General Physics and Astronomy


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