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Insertion Correcting Capability for Quantum Deletion-Correcting Codes

Ken Nakamura, Takayuki Nozaki

TL;DR

This paper proves that any quantum t-deletion-correcting codes also correct a total of t insertion and deletion errors under a certain condition, and proposes the quantum indel distance and describes insertion and deletion errors correcting capability of quantum codes by this distance.

Abstract

This paper proves that any quantum t-deletion-correcting codes also correct a total of t insertion and deletion errors under a certain condition. Here, this condition is that a set of quantum states is defined as a quantum error-correcting code if the error spheres of its states are disjoint, as classical coding theory. In addition, this paper proposes the quantum indel distance and describes insertion and deletion errors correcting capability of quantum codes by this distance.

Insertion Correcting Capability for Quantum Deletion-Correcting Codes

TL;DR

This paper proves that any quantum t-deletion-correcting codes also correct a total of t insertion and deletion errors under a certain condition, and proposes the quantum indel distance and describes insertion and deletion errors correcting capability of quantum codes by this distance.

Abstract

This paper proves that any quantum t-deletion-correcting codes also correct a total of t insertion and deletion errors under a certain condition. Here, this condition is that a set of quantum states is defined as a quantum error-correcting code if the error spheres of its states are disjoint, as classical coding theory. In addition, this paper proposes the quantum indel distance and describes insertion and deletion errors correcting capability of quantum codes by this distance.
Paper Structure (16 sections, 20 theorems, 85 equations)

This paper contains 16 sections, 20 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Quantum States lidar2013quantum
  5. Quantum Errors
  6. Deletion Errors
  7. Insertion Errors
  8. Composite Errors
  9. Quantum States after Insertion Errors to Mixed States
  10. Quantum States after Composite Error
  11. Deletion Correcting Codes Correct Insertion Errors
  12. Properties of Deletion and Insertion Errors
  13. Main Result
  14. Quantum Indel Distance
  15. Definition and Examples
  16. ...and 1 more sections

Key Result

Theorem 1

Consider a quantum state $\rho \in S(\mathbb{C}^{l \otimes n})$ given in Eq. eq:orth_basis_x and a permutation $\tau$ on $[\space[n]\space]$. With some abuse of notation, we define the index permutation$\tau(\rho) \in S(\mathbb{C}^{l \otimes n})$ for $\rho$ by In addition, for $t,n \in \mathbb{Z}^+$ and $Q = \{ q_1, q_2, ..., q_t \} \subset [\space[n+t]\space]$ with $q_1 < q_2 < \cdots < q_t$, l

Theorems & Definitions (42)

  • Theorem 1: sibayama_single_ins_def
  • Example 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • proof
  • Corollary 1
  • Remark 1
  • Corollary 2
  • ...and 32 more