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

TL;DR

This paper establishes a fundamental equivalence between the correctability of deletions and insertions of separable states in quantum codes, providing a unified framework for quantum error correction.

Contribution

It proves that quantum codes correcting deletions automatically correct separable insertions, using an algebraic approach with Kraus operators and the Knill-Laflamme conditions.

Findings

Proved the equivalence between insertion and deletion error correction in quantum codes.

Developed an algebra for commuting Kraus operators representing errors.

Demonstrated that correction of one error type implies correction of the other.

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.