[Paper Review] The equivalence between correctability of deletions and insertions of separable states in quantum codes

Taro Shibayama, Yingkai Ouyang|arXiv (Cornell University)|May 15, 2021

Quantum Computing Algorithms and Architecture22 references13 citations

TL;DR

This paper proves the equivalence between correcting quantum deletion errors and correcting separable insertion errors using the Knill-Laflamme (KL) error-correction conditions. By modeling insertion and deletion channels via Kraus operators and leveraging algebraic manipulation of tensor products, the authors show that any quantum code correcting t deletions automatically corrects t insertions of separable states, establishing a fundamental duality in quantum error correction for these noise types.

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.

Motivation & Objective

• To establish a theoretical equivalence between quantum insertion and deletion error correction capabilities.
• To address the gap in quantum coding theory where insertion error correction has received less attention than deletion correction.
• To unify the treatment of insertion and deletion errors by showing they are operationally equivalent under the Knill-Laflamme framework.
• To extend the applicability of existing deletion-correcting codes to include separable insertion errors.
• To provide a formal algebraic foundation for analyzing insertion/deletion errors using Kraus operators and tensor product rules.

Proposed method

• Formalize quantum insertion and deletion channels using the Kraus operator formalism, representing them as quantum channels with specific operator structures.
• Define (t1, t2)-insdel channels as compositions of t1 insertions and t2 deletions, with insertions of separable states and deletions of unknown qubits.
• Apply the Knill-Laflamme (KL) conditions to derive necessary and sufficient conditions for error correction in terms of inner products of Kraus operators.
• Develop a tensor product calculation framework using lemmas for manipulating Kraus operators across insertion and deletion positions.
• Prove that Kraus operators for (t1, t2)-insdel channels can be transformed into those of equivalent (t1−1, t2+1) or (t1+1, t2−1) channels via algebraic identities.
• Use inductive reasoning based on Lemmas 5.1 and 5.2 to show that if a code corrects t deletions, it also corrects t insertions of separable states, proving Theorem 2.5.

Experimental results

Research questions

• RQ1 Can quantum codes that correct deletion errors also correct insertion errors of separable states?
• RQ2 Is there a fundamental algebraic equivalence between insertion and deletion error correction in quantum codes?
• RQ3 How do the Knill-Laflamme conditions behave under composition of insertion and deletion operations?
• RQ4 Can the Kraus operator formalism be used to unify the analysis of insertion and deletion errors?
• RQ5 What is the role of separable states in insertion errors, and how does their structure affect error correction capability?

Key findings

• Any quantum code that corrects t deletion errors automatically corrects t insertion errors of separable states, establishing a complete equivalence.
• The proof relies on showing that Kraus operators for (t1, t2)-insdel channels can be transformed into those of (t1−1, t2+1) or (t1+1, t2−1) channels via tensor product identities.
• Lemmas 5.1 and 5.2 demonstrate that the error correction capability is preserved when shifting one deletion to an insertion or vice versa.
• The equivalence holds regardless of the specific structure of the inserted separable state, as long as it is a product state over qudits.
• The result implies that permutation-invariant quantum codes, which already correct deletions, also inherently correct separable insertions.
• The framework provides a general method to analyze and construct codes for combined insertion and deletion errors using standard quantum error correction tools.