Relation between discrete and continuous teleportation using linear elements

Abstract

We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation schemes that use only linear optical elements and photodetectors. For this the existing qubit protocols are generalized to qudits with a discrete and finite spectrum but with an arbitrary number of states or alternatively to continuous variables. Correspondingly a generalization of linear optical operations and detection is made. It is shown that linear teleportation is only possible in a probabilistic sense. A general expression for the success probability is derived which is shown to depend only on the dimensions of the input and ancilla Hilbert spaces. From this the known results $p=1/2$ and $p=1$ for the discrete and continuous cases can be recovered. We also discuss the probabilistic teleportation scheme of Knill, Laflamme and Milburn and argue that it does not make optimum use of resources.