We present a unified approach to quantum error correction, called operator quantum error correction. This scheme relies on a generalized notion of noiseless subsystems that is not restricted to the commutant of the interaction algebra. We arrive at the unified approach, which 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, by combining active error correction with this generalized noiseless subsystem method. Moreover, we demonstrate that the quantum error correction condition from the standard model is a necessary condition for all known methods of quantum error correction.PACS numbers: 03.67. Pp, 03.67.Hk, 03.67.Lx The possibility of protecting quantum information against undesirable noise has been a major breakthrough for the field of quantum computing, opening the path to potential practical applications. In this paper, we show that the various techniques used to protect quantum information all fall under the same unified umbrella. First, we will review the standard model for quantum error correction [1,2,3,4], and the passive error prevention methods of "decoherence-free subspaces" [5,6,7,8] and "noiseless subsystems" [9,10,11]. We shall then demonstrate how the latter scheme admits a natural generalization, and study the necessary and sufficient conditions leading to such generalized noiseless subsystems. This generalized method in turn motivates a unified approach -called operator quantum error correction -that incorporates all aforementioned techniques as special cases. We describe this approach and discuss testable conditions that characterize when error correction is possible given a noise model. Moreover, we show that the standard error correction condition is a prerequisite for any of the known forms of error correction/prevention to be feasible.The Standard Model -What could be called the "standard model" for quantum error correction [1, 2, 3, 4] consists of a triple (R, E, C) where C is a subspace, a quantum code, of a Hilbert space H associated with a given quantum system. The error E and recovery R are quantum operations on B(H), the set of operators on H, such that R undoes the effects of E on C in the following sense:where P C is the projector of H onto C. As a prelude to what follows below, let us note that instead of focusing on the subspace C, we could just as easily work with the set of operators B(C) which act on C.When there exists such an R for a given pair E, C, the subspace C is said to be correctable for E. The action of the noise operation E can be described in an operatorsum representation as E(σ) = a E a σE † a . While this representation is not unique, all representations of a given map E are linearly related: if, then there exists scalars u ba such that F b = a u ba E a . We shall identify the map E with any of its error operators E = {E a }. The existence of a recovery operation R of E on C may be cleanly phrased in terms of the {E a } as follows [3,4]:f...

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.Error correction methods are of crucial importance for quantum computing and the so far most general framework, called operator quantum error correction (OQEC) [1,2], encompasses active error correction [4,5,6,7,8] (QEC), together with the concepts of decoherence-free and noiseless subspaces and subsystems [9,10,11,12,13,14,15]. The OQEC approach has enabled more efficient correction procedures in active error correction [16,17,18,19], has led to improved threshold results in fault tolerant quantum computing [20], and has motivated the development of a structure theory for passive error correction [21,22] which has recently been used in quantum gravity [23,24,25,26].In this paper, we introduce a natural generalization of this theory. To this end, we change the focus from that of states to that of observables: conservation of a state by a given noise model implies the conservation of all of its observables, and is therefore a rather strong requirement. This can be alleviated by specifically selecting only some observables to be conserved. In this context it is natural to consider algebras of observables [27]. Hence our codes take the form of operator algebras that are closed under Hermitian conjugation; that is, finite dimensional C * -algebras [28]. As a convenience, we shall simply refer to such operator algebras as "algebras". Correspondingly we refer to the new theory as "operator algebra quantum error correction" (OAQEC). We present results that establish testable conditions for correctability in OAQEC. We also discuss illustrative examples and consider applications to quantum teleportation and information flow in quantum interactions. We shall present the proofs and more examples in [29].Noise models in quantum information are described by channels, which are (in the Schrödinger picture) tracepreserving (TP) and completely positive (CP) linear maps E on mixed states, ρ, which are operators acting on a Hilbert space H. If ρ is a density matrix we can always write ρ → E(ρ) = a E a ρE † a where {E a } is a non-unique family of channel elements. The QEC framework addresses the question of whether a given subspace of states P H, called the code, can be corrected in the sense that there exists a correction channel R such that R(E(ρ)) = ρ for all states ρ in the subspace; that is, all ρ which satisfy ρ = P ρP . This amounts to asking for a subspace on which E has a left inverse that is a ph...