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Modern computing and communication technologies such as supercomputers and the Internet are based on optically connected networks of microwave-frequency information processors. An analogous architecture has been proposed for quantum networks, using optical photons to distribute entanglement between remote superconducting quantum processors. Here we report a step towards such a network by observing non-classical correlations between photons in an optical link and a superconducting quantum device. We generate these states of light through a spontaneous parametric down-conversion process in a chip-scale piezo-optomechanical transducer, and we measure a microwave–optical cross-correlation exceeding the Cauchy–Schwarz classical bound for thermal states. As further evidence of the non-classical character of the microwave–optical photon pairs, we observe antibunching in the microwave state conditioned on detection of an optical photon. Such a transducer can be readily connected to an independent superconducting qubit module and serve as a key building block for optical quantum networks of microwave-frequency qubits.