Quantum state discrimination via linear optics and continuous photon counting

Generalized measurements (POVMs) are important tools in quantum information. So far, there have been extensive analyses to derive optimal POVMs in various contexts. On the other hand, another emerging topic is the implementation of POVMs with restricted physical tools such as linear optics. Recently, general criteria were proposed to decide whether one can implement a given projection measurement via linear optics [1]. In this approach, implementing the projection operators {∏̂_i|π _i〉〈π_i|} is regarded equivalent to the exact discrimination of the orthogonal states {|π_i|〉}. Measurement process can be divided by discrete partial measurement steps and, to perform an exact discrimination, the conditional states after each measurement step must remain orthogonal. The formulae to check whether one can implement such a step via linear optics and photon counting was derived. In this paper, we investigate an expansion of this approach to the limit of continuous photon counting [2]. We discuss the limit that the coupling between the signal and measurement apparatus is infinitesimal at each step and the number of the step is infinitely large. As an example, we consider a specific case where the sates to be discriminated are single-mode states and any nonclassical ancillas are excluded. This scenario is important since if a perfect discrimination is possible in this setup, one can approximate it in practical discrete setup without any expensive ancillas. We show that, only in the continuous limit, one can implement the measurement attaining the minimum error discrimination of binary coherent states {|α〉,|-α〉}, and the projection of photon number qubit signals onto the superposition of zero- and one-photon states {(|0〉±|1〉)/√2} via linear optics without any nonclassical ancillas. The former scheme was firstly proposed by Dolinar (see references in [2]). We will also discuss the implementability of more general binary state discrimination via the same setup.