# Quantum error correction of observables

@article{Bny2007QuantumEC, title={Quantum error correction of observables}, author={C{\'e}dric B{\'e}ny and Achim Kempf and David W. Kribs}, journal={Physical Review A}, year={2007}, volume={76}, pages={042303} }

A formalism for quantum error correction based on operator algebras was introduced by us earlier [Phys. Rev. Lett. 98, 10052 (2007)] via consideration of the Heisenberg picture for quantum dynamics. The resulting theory allows for the correction of hybrid quantum-classical information and does not require an encoded state to be entirely in one of the corresponding subspaces or subsystems. Here, we provide detailed proofs for our earlier results, derive more results, and elucidate key points…

## Figures from this paper

## 44 Citations

Classical Enhancement of Quantum Error-Correcting Codes

- Computer Science, Physics
- 2008

A general formalism for quantum error-correcting codes that encode both classical and quantum information (the EACQ formalism) is presented, which unifies the entanglement-assisted formalism and classical error correction.

ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION

- Computer Science
- 2008

Every hybrid classical-quantum code is shown to be unitarily recoverable in an appropriate sense and the algebraic approach of the proof yields a new, less technical proof for the OQEC case.

Information flow at the quantum-classical boundary

- Computer Science
- 2008

A model of decoherence which can account for the emergence of a classical phase-space is proposed and supports the view that the quantum- classical correspondence is given by a quantum-to-classical channel, i.e. a POVM.

Quantum Error Correction and One-Way LOCC State Distinguishability

- Physics
- 2018

This work considers one-way LOCC measurement protocols as quantum channels and investigates their error correction properties, emphasizing an operator theory approach to the subject, and obtains new applications to one- way LOCC state distinguishability.

Shrinking of operators in quantum error correction and AdS/CFT

- MathematicsJournal of High Energy Physics
- 2019

Abstract
We first show that a class of operators acting on a given bipartite pure state on ℋ
A
⊗ ℋ
B
can shrink its supports on ℋ
A
⊗ ℋ
B
to only ℋ
A
or ℋ
B
while keeping its mappings.…

Quantum error correction on infinite-dimensional Hilbert spaces

- MathematicsJournal of Mathematical Physics
- 2009

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in…

Encoding Classical Information in Gauge Subsystems of Quantum Codes

- Computer ScienceInternational Journal of Quantum Information
- 2022

This paper gives an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code.

The multiplicative domain in quantum error correction

- Computer Science
- 2009

It is shown that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes, which are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes.

Private algebras in quantum information and infinite-dimensional complementarity

- Mathematics
- 2015

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel,…

The Ryu–Takayanagi Formula from Quantum Error Correction

- Mathematics
- 2016

I argue that a version of the quantum-corrected Ryu–Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the…

## References

SHOWING 1-10 OF 32 REFERENCES

Quantum computation and quantum information

- PhysicsMathematical Structures in Computer Science
- 2007

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal…

Decoherence and the Appearance of a Classical World in Quantum Theory

- Physics
- 1996

1 Introduction.- 2 Basic Concepts and Their Interpretation.- 3 Decoherence Through Interaction with the Environment.- 4 Decoherence in Quantum Field Theory and Quantum Gravity.- 5 Consistent…

Mathematical Foundations of Quantum Mechanics

- Physics
- 1955

Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth…

C*-algebras by example

- Mathematics
- 1996

The basics of C*-algebras Normal operators and abelian C*-algebras Approximately finite dimensional (AF) C*-algebras $K$-theory for AF C*-algebras C*-algebras of isometries Irrational rotation…

Inform

- Business
- 1995

Describes an information system for the management of agricultural research, INFORM, developed by the International Service for National Agricultural Research (ISNAR). The system takes data on…

Phys

- Rev. A 55, 900
- 1997

Phys

- Rev. A 52, R2493
- 1995

Phys

- Rev. A 54, 3824
- 1996

Quant

- Inf. & Comp. 6, 382
- 2006