Monotonicity of quantumness of ensembles under commutativity-preserving channels

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A quantum channel (operation) is commutativity-preserving if it maps commutating states to commutating states. We show that commutativity-preserving channels cannot increase quantumness of ensembles, as quantified in terms of the commutator of square roots of constituent states in a quantum ensemble recently introduced [Phys. Rev. A 96, 022132 (2017)].