Necessary and sufficient condition for joint measurability

Abstract

To analyze the joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called W measure, such that it has marginals of positive operator-valued measures. We prove that W measure is a positive operator-valued measure (POVM) if and only if its marginal POVMs are jointly measurable. The proof suggests employing a negative W measure as an indicator of nonjoint measurability. This translates the joint measurability problem to unconstrained convex minimization of nondifferentiable function, which is solvable using a subgradient method. By applying triangle inequalities to the negativity, we derive the analytic joint measurability criteria for dichotomic and trichotomic variables. Moreover, we propose an operational test for joint measurability in a sequential measurement scenario.