Absence of coherent superpositions in the coherent-state basis is a sufficient condition for Wigner function positivity, necessary and sufficient for cat states and limiting cases of higher-order cat states.
Kraus,States, Effects, and Operations: Fundamental Notions of Quantum Theory, vol
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Operational criterion for Wigner function negativity
Absence of coherent superpositions in the coherent-state basis is a sufficient condition for Wigner function positivity, necessary and sufficient for cat states and limiting cases of higher-order cat states.