Probability and fidelity of teleportation in a two-mode continuous variable cluster state via an insufficiently selective measurement

Continuous-variable projective measurements can not select individual measurement results as in the discrete case; instead, the possible outcomes are bounded by the selectivity interval of the measurement; then, it is say that continuous-variable measurement devices are insufficiently selective. By utilizing this concept we show that the probability and fidelity of teleportation in a two-mode cluster state can be handled by the localization of the selectivity interval of the measurement apparatus. Besides, we provide a mathematical expression describing the probability distribution of the measurement outcomes in the two-mode cluster, which is a fundamental solution of the heat equation. In addition, we show that the fidelity of teleportation in the two mode cluster is given by the quotient between the squared solution of a non-homogeneous heat equation and the solution of the conventional heat equation. Furthermore, we extend our approach to a configuration involving successive clusters with intermediate corrections between each teleportation step. To exemplify our proposal, we consider the specific case of a squeezed-coherent state as the quantum state under teleportation.