Measurement operators for quantum coherent states in binary and quaternary photonic communications with polarization degree of freedom

The objective of this work is the derivation of the measurement operators and the error probability performance in the quantum detection of information-carrying coherent states used in binary and quaternary 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 four-dimensional (4D) modulation in the form of a constellation of quantum composite coherent states to be transmitted in the optical channel. At the receiver, measurements are performed to determine as precisely as possible which state was sent, with a detection/discrimination strategy that must allow: (a) the derivation of the quantum measurement operators for the considered constellation; and (b) the evaluation of the system performance in terms of bit error probability. With this purpose, in this work we apply the quantum square root method (SRM) for the detection/discrimination strategy on the received 4D binary and quaternary constellations, corresponding to those commonly used in classical photonic communications such as: (a) polarization shift-keying, (b) polarization switching, and (c) polarization multiplexing (PM), but now with the 4D composite quantum states transmitted in the photonic channel. For this task, based on the inner-product Gram matrices for the different constellations studied, we apply the SRM method to obtain the measurement operators as linear combinations of the generating composite quantum coherent states. We analytically derive the measurement operators and their behavior as a function of the photon number per symbol, for the 4D modulation formats analyzed, and we present graphical descriptions in their quadrature/polarization representations. We evaluate the bit error probability performance from a built-in functionality of the SRM method, that allows its derivation from the weighted inner products between the normalized measurement operators and their corresponding generating composite coherent states. Finally, we suggest transmitter–receiver configurations for some of these quantum coherent states 4D constellations.