Coherence Equality and Communication in a Quantum Superposition

In this Letter, we introduce a coherence equality that is satisfied by any classical communication-i.e., conveyed by a localized carrier traveling along well defined directions. In contrast, this equality is violated when the carrier is prepared in coherent quantum superposition of communication directions. This is phrased in terms of the success probability of a certain communication task, which is always constant and equal to 1= 2 in the classical case. On the other hand, we develop two simple quantum schemes that deviate systematically from the classical value, thus, violating the coherence equality. Such a violation can also be exploited as an operational way to witness spatial quantum superpositions without requiring us to recombine the modes in a standard interferometer, but only by means of spatially separated local measurements.