Asymptotic linear scaling of the optimal coherence scale

Prove or disprove the conjectured asymptotic law \(\mathcal C(N)\sim N\) for the minimum quadrature coherence scale among quantum states with Wigner negativity \(N\) as \(N\to\infty\).

Background

For Fock states and optimal cubic Gaussian states, the paper finds that the quadrature coherence scale grows linearly, up to asymptotic constants, with Wigner negativity. The authors propose that this scaling may hold for the globally optimized quantity C(N)\mathcal C(N).

Establishing this stronger asymptotic statement would sharpen the divergence conjecture and would imply a universal quadratic upper bound on decoherence time in terms of Wigner negativity.

References

More precisely, one may expect that, as for the Fock states and the optimal cubic Gaussian states \begin{equation} \mathcal C(N)\sim N, \end{equation} for large $N$.

— Tradeoff between Wigner negativity and decoherence time for cubic Gaussian states  (2610.10073 - Arnhem et al., 7 Oct 2026) in Section Conclusion and outlook