Universal divergence of the minimum quadrature coherence scale at large Wigner negativity

Prove that the minimum quadrature coherence scale among all quantum states with fixed Wigner negativity diverges as the negativity tends to infinity; specifically, establish that, for \(\mathcal C(N)=\min_{\rho,\,\mathcal N(\rho)=N}\mathcal C(\rho)\), one has \(\lim_{N\to+\infty}\mathcal C(N)=+\infty\).

Background

The authors define C(N)\mathcal C(N) as the smallest quadrature coherence scale achievable by any state whose Wigner negativity equals NN. Since decoherence time is inversely related to the square of the quadrature coherence scale, divergence of C(N)\mathcal C(N) would imply that sufficiently negative states necessarily have arbitrarily short decoherence times.

The paper establishes this behavior for cubic Gaussian states and observes the same qualitative tradeoff for Fock states, but leaves the corresponding statement for the full set of quantum states as a conjecture.

References

More generally, we formulate the following conjecture. Let \begin{equation} \mathcal C(N)=\min_{\rho, \mathcal N(\rho)=N} \mathcal C(\rho) ; \end{equation} this is the smallest quadrature coherence scale (and hence the largest coherence time) of any state with a fixed Wigner negativity $N$. Then \begin{equation} \lim_{N\to+\infty} \mathcal C(N)=+\infty. \end{equation}

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