Probability and fidelity of teleportation in a two-mode continuous-variable cluster state via a finite-resolution measurement device

Abstract Continuous-variable measurements cannot select individual outputs as in the discrete case; instead, the possible results are determined with a finite resolution. Then, it is said 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 continuous-variable cluster state can be handled by both the localization and width of the selectivity interval of the measurement apparatus. Furthermore, we identify a trade-off relationship between the probability and fidelity of teleportation, which depends on both the width of the selectivity interval and the level of squeezing achieved in the cluster. Besides, we provide the mathematical expression for the probability distribution associated with the likelihood of teleportation 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 the quotient between the squared solution of a non-homogeneous heat equation and the solution of the conventional heat equation. 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.