A receiver for quadrature/polarization modulated quantum coherent states in photonic communications employing the Naimark extension

This work deals with the derivation of a receiver structure for the quantum detection of information-carrying coherent states used in photonic communications systems, that employ the degrees of freedom of both the complex amplitude and the state of polarization in the quantum state, producing a high dimensional modulation in the form of a constellation of quantum composite coherent states to be transmitted in the optical channel. Our receiver is based on the extension of the positive operator value measurements (POVM) in the constellation Hilbert space, towards projective measurements in a larger Hilbert space. Starting from the measurement vectors obtained from an optimum detection/discrimination strategy-the square root method (SRM) in our case-, and basing our analysis on the Naimark extension, we present a procedure for building those projectors for a received modulation format of composite coherent states. We analytically derive the orthogonal and idempotent projectors, arriving at closer form expressions for the considered modulation format, their decomposition in unitary rotations, and suggest a receiver configuration for its physical realization. Finally, an error probability analysis of our modulation formats is presented.