Properties of multi-moded phase-randomized coherent states

Phase-randomized coherent states are widely used in various applications of quantum optics. They are best known to be the core part of decoy-state quantum key distribution protocols with phase-coding. From the perspective of future development of quantum protocol architecture, it is important to determine whether phase randomization can be applied at an arbitrary stage of an optical scheme without affecting the informational properties of the quantum system. In this paper, using the superoperator formalism, we have shown that phase randomization of a two-mode coherent state commutes with linear optical transformations. This implies that phase randomization can be applied virtually at any point within the optical setup. We further demonstrate that the Holevo bound for such a state coincides with that of regular coherent states, bearing in mind that the Holevo bound quantifies only the maximum amount of information accessible to an eavesdropper. Advantages of phase-randomized coherent states compare to regular ones in particular cases of eavesdropper’s strategies should be considered separately. Also, these findings indicate that phase randomization can be directly applied to a subcarrier wave quantum key distribution type of systems, opening prospects for its future development.