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arXiv:0705.1574 · Physical Review A

Quantum error correction of observables

Cédric Bény, Achim Kempf, David W. Kribs

Published Oct 2, 2007• 67 citations
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Abstract

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.

DOI
10.1103/physreva.76.042303

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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.
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.
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.
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error-correcting codes, one requires the ability to apply a procedure which exactly reverses on the error-correcting subspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform corrections only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error-correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error-correction procedures.
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''.