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A parameter whose coupling to a quantum probe of n constituents includes all two-body interactions between the constituents can be measured with an uncertainty that scales as 1/n to the 3/2, even when the constituents are initially unentangled. We devise a protocol that achieves the 1/n to the 3/2 scaling without generating any entanglement among the constituents, and we suggest that the protocol might be implemented in a two-component {Bose-Einstein} condensate.