Hilbert’s 17th Problem and the Quantumness of States

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.

Sudden vanishing and reappearance of nonclassical effects: General occurrence of finite-time decays and periodic vanishings of nonclassicality and entanglement witnesses

“…By analogy with definitions of entanglement witness (see the following section), the normallyordered Hermitian operator :f †f : can be referred to as (nonlinear) nonclassicality (or quantumness) witness [14]. For convenience, we call the nonclassicality witness (and also entanglement witness) not only an observable but also its expectation value.…”

“…Here we apply the method of constructing nonclassicality witnesses proposed in Refs. [13,14] and developed in Refs. [19,25].…”

“…Then one can apply another criterion of nonclassicality [13,14]: These conditions can be compactly written as :f †f : cl ≥ 0 and :f †f : ncl < 0. By analogy with definitions of entanglement witness (see the following section), the normallyordered Hermitian operator :f †f : can be referred to as (nonlinear) nonclassicality (or quantumness) witness [14].…”

“…Namely, the SV and SR effects can also be observed during the evolution of entanglement witnesses [9][10][11] (for a review see Ref. [12]) and nonclassicality witnesses (also called quantumness witnesses) [13][14][15][16][17][18][19] corresponding to violations of classical inequalities.…”

Collapse and revival of quantum coherence for a harmonic oscillator interacting with a classical fluctuating environment

“…In this context, a thoroughly studied quantum system is the single-mode quantum harmonic oscillator interacting with a bosonic bath of oscillators. For such an open system, the decoherence time, ruling the transition from the quantum to the classical regime, may be identified by different nonclassicality criteria, which have been widely investigated [11][12][13][14][15][16][17][18][19][20][21][22][23][24] and compared [25]. Extensions to multimode systems [26][27][28][29] have been analyzed and the decoherence process has been addressed extensively [30,31].…”