The equivalence between correctability of deletions and insertions of separable states in quantum codes

Abstract

In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.

Linear programming bounds for quantum amplitude damping codes

“…Such codes have been studied in the qubit [22], [23], [20], [24], the qudit [25], and the bosonic [26] settings. These quantum codes are interesting because of not only their capability to correct non-trivial errors such as quantum deletions [27], [28] and insertions [29], but also their potential applications as quantum memories [30] and for robust quantum metrology [31]. One key attractive feature of permutation-invariant quantum codes is the ease in which they can be prepared in physical systems [32], [33] as compared to the usual stabilizer codes.…”