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Operator quantum error correction

David W. Kribs, Raymond Laflamme, David Poulin, Maia Lesosky

TL;DR

<3-5 sentence high-level summary>

Abstract

This paper is an expanded and more detailed version of our recent work in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques - i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method - as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of ``unitarily noiseless subsystems''.

Operator quantum error correction

TL;DR

<3-5 sentence high-level summary>

Abstract

This paper is an expanded and more detailed version of our recent work in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques - i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method - as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of ``unitarily noiseless subsystems''.

Paper Structure

This paper contains 8 sections, 10 theorems, 81 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Quantum Operations
  3. Standard Model for Quantum Error Correction
  4. Noiseless Subsystems and Decoherence-Free Subspaces
  5. Noiseless Subsystems For Arbitrary Quantum Operations
  6. Operator Quantum Error Correction
  7. Unitarily Noiseless Subsystems
  8. Conclusion

Key Result

Lemma 2.1

The map $\Gamma : {\mathcal{B}}({\mathcal{H}})\rightarrow{\mathcal{B}}({\mathcal{H}})$ given by $\Gamma = \{P_{kl}\}$ satisfies the following: for all operators $\sigma \in {\mathcal{B}}({\mathcal{H}})$, so in particular $\Gamma(\sigma^A\otimes\sigma^B) \propto {\rm 1 l}^A\otimes \sigma^B$ for all $\sigma^A$ and $\sigma^B$.

Figures (1)

  • Figure 1:

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 11 more