Conditions for the Approximate Correction of Algebras

We study the approximate correctability of general algebras of observables, which represent hybrid quantum-classical information. This includes approximate quantum error correcting codes and subsystems codes. We show that the main result of [1] yields a natural generalization of the Knill-Laflamme conditions in the form of a dimension independent estimate of the optimal reconstruction error for a given encoding, measured using the trace-norm distance to a noiseless channel. Slightly relaxing the requirement of perfect quantum error correction can allow for significantly larger quantum codes [2,3]. Here we focus on a quantification of the correction error based on the diamond norm distance, introduced below, which can be related to the worst case entanglement fidelity. (See [4] for the case of average entanglement fidelity). There exists results giving sufficient conditions for a code to be approximately correctable in that sense [5,6], however it is not known how general these conditions are. Instead we want to draw attention to results by Kretschmann et al. [1,7] who gave lower and upper bounds for the optimal reconstruction error for a given code in terms of the complementary channel’s distance to a maximally forgetful channel. The present report can be seen partly as an advertisement of these results in a context where they are not widely known, or their meaning not recognized, namely as a providing a necessary and sufficient condition for approximate error correction. In addition we improve on these results by rendering the conditions more explicit, and generalizing them to the correction of general algebras. The condition that we obtain (Theorem 1) can be understood as a perturbation of the exact Knill-Laflamme condition [8], or more generally its subsystem version [9], or full algebraic form [10]. We also give an essentially equivalent condition based on individual observables of the algebra (Theorem 2). The correctable algebra can be understood as representing a quantum system with superselection rules, or a hybrid quantum-classical memory [11], and can be shown to be the most general type of exactly correctable information in the sense of [12].