Faithful quantumness certification for single-mode continuous-variable systems

Develop a quantumness certification functional for one-dimensional continuous-variable (single-mode) quantum states that is faithfully discriminating, meaning it always detects and certifies every nonclassical state and certifies only classical states as classical. In phase-space terms, ensure the functional attains negative values somewhere for every nonclassical state and remains nonnegative for all classical states, providing a universally applicable method for both theoretical and experimentally reconstructed states.

Background

The paper studies phase-space-based certification of nonclassicality for one-dimensional continuous-variable systems. While the Sudarshan-Glauber P distribution provides a definition of nonclassicality via negativity, it is typically too singular to reconstruct or analyze in practice. Bohmann and Agudelo’s functional xi(x,p) was shown to be sufficient: xi(x,p) < 0 implies nonclassicality, but it is not necessary, leading to failures on weakly nonclassical states.

This work generalizes xi to a new family of certification functionals S(T,ΔT) that can be more sensitive, yet both xi and S fail for very weakly nonclassical states. Consequently, a universally faithful certification—one that necessarily detects all nonclassical states and never mislabels them as classical—remains unresolved.

References

Yet, they also fail for very weakly nonclassical states, in other words, the question how to faithfully certify quantumness remains an open question.

— Towards a Faithful Quantumness Certification Functional for One-Dimensional Continuous-Variable Systems  (2512.23299 - Steuernagel et al., 29 Dec 2025) in Abstract (page 1)

It remains unclear how this can inform any quantum signals that can be measured in the $x$ basis alone, and whether this perspective is distinct from the SQU test.

— For Whom Does Bell Hold?  (2609.10535 - Green et al., 9 Sep 2026) in Section 5.2, “Negativity of the Wigner Distribution”

Yet it is readily shown that the onset of quantumness signalled by an uncertainty below the classical floor---$\Delta x<1/2$ or $\Delta p<1/2$---already implies $P(\alpha)<0$ for squeezed thermal states. Whether the Sudarshan--Glauber $P$-function of such a state is negative or singular is thus left unclear.

— How Quantum Is Light? Concepts, Measures, and Applications of Optical Quantumness  (2610.25894 - Bose et al., 19 Oct 2026) in Section 6, “Negativity vs singularity and extremal states”

How contextuality relates to non-regularity of the $P$-function remains obscure, largely because of the mathematical intractability of the $P$-function itself.

— How Quantum Is Light? Concepts, Measures, and Applications of Optical Quantumness  (2610.25894 - Bose et al., 19 Oct 2026) in Section 6, “Does quantumness imply contextuality?”