Revealing nonclassicality beyond Gaussian states via a single marginal distribution

Abstract

A standard method to obtain information on a quantum state is to measure marginal distributions along many different axes in phase space, which forms a basis of quantum-state tomography. We theoretically propose and experimentally demonstrate a general framework to manifest nonclassicality by observing a single marginal distribution only, which provides a unique insight into nonclassicality and a practical applicability to various quantum systems. Our approach maps the 1D marginal distribution into a factorized 2D distribution by multiplying the measured distribution or the vacuum-state distribution along an orthogonal axis. The resulting fictitious Wigner function becomes unphysical only for a nonclassical state; thus the negativity of the corresponding density operator provides evidence of nonclassicality. Furthermore, the negativity measured this way yields a lower bound for entanglement potential-a measure of entanglement generated using a nonclassical state with a beam-splitter setting that is a prototypical model to produce continuous-variable (CV) entangled states. Our approach detects both Gaussian and non-Gaussian nonclassical states in a reliable and efficient manner. Remarkably, it works regardless of measurement axis for all non-Gaussian states in finite-dimensional Fock space of any size, also extending to infinite-dimensional states of experimental relevance for CV quantum informatics. We experimentally illustrate the power of our criterion for motional states of a trapped ion, confirming their nonclassicality in a measurementaxis-independent manner. We also address an extension of our approach combined with phase-shift operations, which leads to a stronger test of nonclassicality, that is, detection of genuine nonGaussianity under a CV measurement.N onclassicality is a fundamentally profound concept to identify quantum phenomena inaccessible from classical physics. It also provides a practically useful resource, for example, entanglement, making possible a lot of applications in quantum information processing beyond classical counterparts (1-3). A wide range of quantum systems, for example, field amplitudes of light, collective spins of atomic ensembles, motional modes of trapped ions, and Bose-Einstein condensate and mechanical oscillators, can be used for quantum information processing based on continuous variables (CVs) (2). It is of crucial importance to establish efficient and reliable criteria of nonclassicality for CV systems, desirably testable with fewer experimental resources, for example, fewer measurement settings (4-8) and with the capability of detecting a broad class of nonclassical states. In this paper, in view of the Glauber-Sudarshan P function (9, 10), those states that cannot be represented as a convex mixture of coherent states are referred to as nonclassical.A standard method to obtain information on a CV quantum state is to measure marginal distributions along many different axes in phase space constituting quantum-state tomography (11).This tomographic reconstruction may r...

Sufficient condition for a quantum state to be genuinely quantum non-Gaussian

“…More recently, Park et al established a Bell-type test in phase space to address nonclassicality and quantum non-Gaussianity for single-mode fields [17], while another efficient formalism was also developed to adopt the marginal distributions of the Wigner function [18].…”

Entropic nonclassicality and quantum non-Gaussianity tests via beam splitting

“…Remark : We note that the entropic test in Eq. (8) can be considered as a subset of our previously proposed criteria, i.e., the so-called demarginalization map (DM) approach 53 . In DM method, nonclassicality is confirmed by showing the unphysicality of a fictitious Wigner function, e.g., constructed as where is the quadrature distribution of a given state and that of a vacuum state.…”

Optomechanical state reconstruction and nonclassicality verification beyond the resolved-sideband regime

“…To verify the legitimacy of the quantum state corresponding to the above fictitious quasiprobability distributions, one can simply employ all the arguments provided in Ref. [28] by using appropriate quasiprobability distributions to compute the matrix elements of the fictitious density operator using the phase-space trace relation. This shows that our method can be used to verify P-function nonclassicality even in the regime of weak interactions.…”

“…(25) and (26) To check the legitimacy of the fictitious density operators resulting from these demarginalization maps, one may follow the standard arguments provided in Ref. [28], such as the Kastler-Loupias-Miracle-Sole test [59][60][61]. In the general formalism, to perform such tests one may also require evaluation of various Fock basis elements of the fictitious density matrix.…”