  The discrimination of coherent states is a key task in optical communication
and quantum key distribution protocols. In this work, we use a
photon-number-resolving detector, the transition-edge sensor, to discriminate
binary-phase-shifted coherent states at a telecom wavelength. Owing to its
dynamic range and high efficiency, we achieve a bit error probability that
unconditionally exceeds the standard quantum limit (SQL) by up to 7.7 dB. The
improvement to the SQL persists for signals containing up to approximately
seven photons on average and is achieved in a single shot (i.e. without
measurement feedback), thus making our approach compatible with larger
bandwidths. Our work shows that a quantum technology can provide a substantial
advantage over conventional technologies in a task of practical importance.
