ar X iv : q ua nt - p h / 05 04 18 9 v 3 17 J an 2 00 6 OPERATOR QUANTUM ERROR CORRECTION

This paper is an expanded and more detailed version of the work [1] 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”. A unified and generalized approach to quantum error correction, called Operator Quantum Error Correction (OQEC), was recently introduced in [1]. This formalism unifies all of the known techniques for the error correction of quantum operations – i.e. the standard model [2, 3, 4, 5], the method of decoherence-free subspaces [6, 7, 8, 9] and the noiseless subsystem method [10, 11, 12] – under a single umbrella. An important new framework introduced as part of this scheme opens up the possibility of studying noiseless subsystems for arbitrary quantum operations. This paper is an expanded and more detailed version of the work [1]. We provide complete details for proofs sketched there, and in some cases we present an alternative “operator” approach that leads to new information. Specifically, we show that correction of the general codes introduced in [1] is equivalent to correction of certain operator algebras, and we use this to give a new proof for the main testable conditions in this scheme. In addition, we discuss a number of examples throughout the paper, and introduce the notion of “unitarily noiseless subsystems” as a relaxation of the requirement in the noiseless subsystem formalism for immunity to errors. 1. Preliminaries 1.1. Quantum Operations. Let H be a (finite-dimensional) Hilbert space and let B(H) be the set of operators on H. A quantum operation (or channel, or evolution) on H is a linear map E : B(H) → B(H) 1 2 D.W. KRIBS, R. LAFLAMME, D. POULIN, M. LESOSKY that is completely positive and preserves traces. Every channel has an “operator-sum representation” of the form E(σ) = ∑aEaσE a, ∀σ ∈ B(H), where {Ea} ⊆ B(H) are the Kraus operators (or errors) associated with E . As a convenience we shall write E = {Ea} when the Ea determine E in this way. The choice of operators that yield this form is not unique, but if E = {Ea} = {Fb} (without loss of generality assume the cardinalities of the sets are the same), then there is a unitary matrix U = (uab) such that Ea = ∑ b uabFb ∀ a. The map E is said to be unital or bistochastic if E(1l) = ∑ aEaE † a = 1l. Trace preservation of E can be phrased in terms of the error operators via the equation ∑ aE † aEa = 1l, which is equivalent to the dual map for E being unital. 1.2. Standard Model for Quantum Error Correction. The “Standard Model” for the error correction of quantum operations [2, 3, 4, 5] consists of triples (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) such that R undoes the effects of E on C in the following sense: (R ◦ E) (σ) = σ ∀σ = PCσPC, (1) where PC is the projection of H onto C. When there exists such an R for a given pair E , C, the subspace C is said to be correctable for E . The existence of a recovery operation R of E = {Ea} on C may be cleanly phrased in terms of the {Ea} as follows [4, 5]: (2) PCE † aEbPC = λabPC ∀ a, b for some matrix Λ = (λab). It is easy to see that this condition is independent of the operator-sum representation for E . 1.3. Noiseless Subsystems and Decoherence-Free Subspaces. Let E = {Ea} be a quantum operation on H. Let A be the C∗-algebra generated by the Ea, soA = Alg{Ea, E† a}. This is the set of polynomials in the Ea and E † a. As a †-algebra (i.e., a finite-dimensional C∗-algebra [13, 14, 15]), A has a unique decomposition up to unitary equivalence of the form A ∼= ⊕ J ( MmJ ⊗ 1lnJ )