Testing nonclassicality in multimode fields: A unified derivation of classical inequalities

Abstract

We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode P functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two-and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.

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

“…But, in fact, the singularity of the P function is implied by its nonpositivity (see, e.g., Ref. [19]). …”

“…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.…”

Quantum-entangled light from localized emitters

“…Note that NPT entanglement is an important class of quantum entanglement, which is useful for many applications [21]. Based on the methods in [15,16], the relation between nonclassicality and entanglement conditions has been discussed in some detail [22]. * Electronic address: peter.gruenwald@uni-rostock.…”

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

“…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]. Besides the fundamental interest, the analysis of the quantum-to-classical transition has relevant applications in the field of quantum technology.…”