Measurement incompatibility and the negativity of quasiprobability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill (MH) quasiprobability distribution associated with two dichotomic observables. We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension $d$. We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension $d$, the positivity of the MH distribution is equivalent to joint measurability, i.e., the MH distribution is positive if and only if the measurements are jointly measurable. We further introduce the MH-like quasiprobability distribution for $n$ dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasiprobability which in turn provides a test of incompatibility.

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• Quantum foundations
• Quantum measurements