Generalized Multiplicative Domains and Quantum Error Correction

Abstract

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.

14 pages

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• Spectral Properties of Tensor Products of Channels
• Quantum Complementarity and Operator Structures
• 2-positive almost order zero maps and decomposition rank
• A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement
• Nullspaces of Entanglement Breaking Channels and Applications