Necessary and sufficient condition for constructing a single qudit insertion/deletion code and its decoding algorithm

Abstract: This paper shows that Knill-Laflamme condition, known as a necessary and sufficient condition for quantum error-correction, can be applied to quantum errors where the number of particles changes before and after the error. This fact shows that correctabilities of single deletion errors and single insertion errors are equivalent. By applying Knill-Laflamme condition, we generalize the previously known correction conditions for single insertion and deletion errors to necessary and sufficient level. By giving an example that satisfies this condition, we construct a new single qudit insertion/deletion code and explain its decoding algorithm.