For energy-bounded bosonic states, a pure state is non-Gaussian exactly when its complex-extended position wavefunction has at least one zero, and the number of zeros equals the stellar rank.
Everett, The theory of the universal wave function, in The Many Worlds Interpretation of Quantum Mechan- ics, edited by B
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On the complex zeros of the wavefunction
For energy-bounded bosonic states, a pure state is non-Gaussian exactly when its complex-extended position wavefunction has at least one zero, and the number of zeros equals the stellar rank.