Implementing nonprojective measurements via linear optics: An approach based on optimal quantum-state discrimination

We discuss the problem of implementing generalized measurements [positive operator-valued measures (POVMs)] with linear optics, either based upon a static linear array or including conditional dynamics. In our approach, a given POVM shall be identified as a solution to an optimization problem for a chosen cost function. We formulate a general principle: the implementation is only possible if a linear-optics circuit exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing the corresponding quantum states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics implementation of the POVM that realizes the quantum mechanically optimal unambiguous discrimination of two pure nonorthogonal states. This extends our previous results on projection measurements and the exact discrimination of orthogonal states.

Figure 1

(Color online) An equivalent description of the detection mechanism after a unitary state transformation via dephasing. In the case of photon counting, the dephased density matrix is a mixture of all possible photon number patterns for a given input state and a given state transformation. Here, we are mainly concerned about linear-optics transformations.

Figure 2

Implementing the optimal unambiguous discrimination of two symmetric coherent states via a simple 50–50 beam splitter and an auxiliary coherent state of the same amplitude.