Classical capacity of unspeakable phase information via quantum systems

Reference frame alignment, such as synchronization of clocks or alignment of an orthogonal triple of axis, is the communication of unspeakable information and requires physical systems with particular degrees of freedom, such as optical phase or angular momentum. An important property of alignment protocols is the ability to reliably and efficiently transmit such unspeakable information. We derive the optimal rate of transmission of information for the case where two parties wish to align their respective local phase references via the exchange of photons. The rate of transmission depends on the variance of the quantum mechanical state used to encode phase information, and can be achieved by performing collective measurements on N photons.