Towards a Faithful Quantumness Certification Functional for One-Dimensional Continuous-Variable Systems

TL;DR

This paper examines the limitations of a recently proposed phase space-based functional for certifying nonclassicality in quantum states, showing it can fail for weakly nonclassical states and proposing generalized, more sensitive functions.

Contribution

It identifies the failure cases of existing certification functionals and introduces generalized forms to improve sensitivity in detecting nonclassicality.

Findings

Existing certification functional can be non-negative for nonclassical states.

Generalized certification functions offer improved sensitivity.

Certifying weakly nonclassical states remains challenging.

Abstract

If the phase space-based Glauber-Sudarshan distribution, $P_{\rho}$, has negative values the quantum state,~$\rho$, it describes is nonclassical. Due to $P$'s singular behaviour this simple criterion is impractical to use. Recent work [Bohmann and Agudelo, Phys. Rev. Lett. 124, 133601 (2020)] presented a general, sensitive, and noise-tolerant certification functional,~$\xi[P]$, for the detection of non-classical behaviour of quantum states $P_{\rho}$. There, it was shown that when this functional takes on negative values somewhere in phase space,~$\xi[P](x,p) < 0$, this is \emph{sufficient} to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification functions is non-negative, $\xi(x,p) \geq 0$, everywhere in phase space. We generalize $\xi$ giving it an appealing form which allows for improved certification. This way we generate a more sensitive family of certification functions. Yet, also these fail for very weakly nonclassical states, the question how to faithfully certify quantumness remains an open question.