Trading energy for coherence in mode discrimination

Energy is arguably the main resource in classical physics; coherence is the prototypical quantum resource. In this paper, we study the interplay between these two resources in the task of optimal mode discrimination. This task, motivated by cryptography and metrology, consists of finding in which mode a signal has been encoded, for a given finite set of possible modes. After recalling that the task can be cast as a semi-definite program, we solve completely the case of two-mode discrimination under a hard energy constraint $n\le N$. Specifically, we identify the region of modes overlap, for which the maximal energy state (the Fock state with N excitations) is the best choice for discrimination. For other overlaps, having access to coherence leads to better discrimination.