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.

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

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.

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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 -