Extremal P-function behavior at fixed energy

Determine which is extremal at a given mean average energy: a negative P-function or a singular P-function.

Background

The review distinguishes two ways in which the Glauber–Sudarshan representation can depart from a classical probability distribution: it may become negative or develop singularities. Existing work has explored relations between these behaviors, but does not settle their relative extremality.

The open problem asks how negative and singular P-functions should be compared for states constrained by the same mean energy, potentially providing an ordering of different forms of optical quantumness.

References

This further opens up the question of which one is extremal between the negative-$P$ and singular-$P$ functions, at a given mean average energy. While sheds some light on the relation between the negativity and the singularity of the $P$-function, it is yet to be settled.

— 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”