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Generalization of Quantum Error Correction via the Heisenberg Picture and Application to Information Flow

Cédric Bény, Achim Kempf, David W. Kribs

Published Aug 8, 2006• 1 citation
OpenAlex

Abstract

We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables and may therefore suitably be called "operator algebra quantum error correction". It allows for the simultaneous correction of several subsystems and does not require the state to be entirely in one of the corresponding subspaces. We also show how this formulation offers a formal framework for the study of information flows in quantum interactions, with applications to decoherence and quantum measurements.

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We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables and may therefore suitably be called "operator algebra quantum error correction". It allows for the simultaneous correction of several subsystems and does not require the state to be entirely in one of the corresponding subspaces. We also show how this formulation offers a formal framework for the study of information flows in quantum interactions, with applications to decoherence and quantum measurements.
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called "operator algebra quantum error correction"). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of information flows in quantum interactions.
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 with expanded discussions. We also present several examples and indicate how the theory can be extended to operator spaces and general positive operator-valued measures.
Operator quantum error correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error correction, and provides a unified framework for topics such as quantum error correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions that characterize when operator quantum error correction is feasible; (b) a representation theorem for a class of noise processes that can be corrected using operator quantum error correction; and (c) generalizations of the coherent information and quantum data processing inequality to the setting of operator quantum error correction.
A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.
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''.
This paper is an expanded and more detailed version of the work \cite{KLP04} 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''.