Probing nonclassicality with matrices of phase-space distributions

Abstract

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyond s-parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

“…And where the equations were specifically states for qudit states, general expressions can be found to calculate these values for structures of even continuous‐variable systems. Further to these measures there are other ways in which phase‐space methods have been useful to characterize a quantum state, these are: measures of coherence [ 100,101 ] ; phase‐space inequalities [ 102–105 ] ; and the negativity in the Wigner function, which plays an important role. The manifestation of these negative values provide nuanced details for each system.…”

Identifying nonclassicality from experimental data using artificial neural networks

“…A different approach is nonclassicality certification via negativities of reconstructed quasiprobabilities [23] (in particular, the Glauber-Sudarshan P function [24] and the Wigner function [25][26][27][28]). Methods that involve regularizations of quasiprobabilities have been implemented for the singlemode and multimode scenarios [29,30], and more recently, phase-space inequalities have been proposed and tested experimentally [31][32][33]. In order to guarantee for negativities with a high statistical significance, the collected experimental data has to be postprocessed after the experiment.…”

Nonclassical kernels in continuous-variable systems

“…In this paper, building on a recently proposed phasespace inequality [15][16][17], we drive a nonclassicality witness for kernel functions. We use our proposed witness * farid.ghobadi80@gmail.com to show that the nonclassical kernel leads to better separation of data points in feature space.…”

“…Instead of working with P distribution, it is more convenient to work with quasiprobability distributions such as the Wigner function, which are non-singular and experimentally accessible [25]. In this paper, we are especially interested in the following inequality, which holds for classical states [15,16]…”

“…( 7) for α = α signals the nonclassicality of the state. Notably, (7) can capture the quantum nature of states with non-negative Wigner functions [15,16].…”