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

TL;DR

The paper tackles faithful quantumness certification in one-dimensional continuous-variable systems using phase-space methods. It extends the Bohmann–Agudelo quantumness functional $\xi(x,p)$ by introducing a smoothing-based family ${\cal S}(T,\Delta T)$, which reduces to $\xi$ when $\Delta T=T$ and connects to $S$-parametrized distributions, with coherent states yielding ${\cal S}=0$. It shows that ${\cal S}$ can be more discriminating than $\xi$, particularly under experimental smearing with $T\ge 1$, but provides explicit counterexamples where both fail for very weak nonclassicality, thereby confirming that universal faithfulness remains an open challenge. Overall, the work offers the most sensitive family of phase-space quantumness functionals to date while clarifying fundamental limits of phase-space-based nonclassicality certification and highlighting the ongoing need for faithfully discriminating criteria.

Abstract

If the phase space-based Sudarshan-Glauber distribution, $P_ρ$, has negative values the quantum state, $ρ$, it describes is nonclassical. Due to $P$'s singular behavior 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, $ξ_{P}$, for the detection of non-classical behavior of quantum states $P_ρ$. There, it was shown that when this functional takes on negative values somewhere in phase space, $ξ_{P}(x,p) < 0$, 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 positive $ξ(x,p) \geq 0$ everywhere in phase space. We generalize $ξ$ giving it an appealing form which allows for improved certification. This way we generate the best family of certification functions available so far. Yet, they also fail for very weakly nonclassical states, in other words, the question how to faithfully certify quantumness remains an open question.

Figures (1)

• Figure 1: Here we study $\xi(T=1)$ and ${\cal S}(T=1,\Delta T)$ for Wigner's distribution, ${\cal P}(x,p;T=1)$, since it is the borderline case experimentally accessible in state reconstruction (see remarks after Eq. (\ref{['eq:SmoothingMeasure2']})). We consider the weakly nonclassical state (\ref{['eq:_BorderCase_state']}): for small values of $w_1 \leq 0.0125$. The certification functional $\xi(x,p;1)$ of Ref. Bohmann_PRL20 (red band) fails to drop to nega-tive values (below green zero-value sheets), i.e., it fails to detect the nonclassical contribution $w_1 W_1$ in state (\ref{['eq:_BorderCase_state']}) for the value $w_1 = 0.0122$ (bottom panels), whereas the certification functional ${\cal S}(x,p ;T=1,\Delta T)$ of Eq. (\ref{['eq:SmoothingMeasure']}) still forms negative values. For slightly smaller values of $w_1 < 0.0122$ (not shown) ${\cal S}$ fails as well. In short, ${\cal S}$ can be more discriminating than $\xi$ but not faithfully so.