Generalized multiplicative domains and quantum error correction

Abstract

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 representationtheoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherencefree subspaces.

Private algebras in quantum information and infinite-dimensional complementarity

“…This in turn leads to a more general notion of complementary channel, and we establish a generalized complementarity theorem for arbitrary dimensions in the new framework. As a corollary, we also obtain a structure theorem for correctable subalgebras that generalizes a finite-dimensional result [20]. We finish by illustrating the framework and concepts for infinite-dimensional linear bosonic channels and a specific class of Gaussian quantum channels [15,17].…”

“…We next present an application of Theorem 4.7 by generalizing the main result in [20] concerning the structure of correctable subsystems for finitedimensional channels as generalized multiplicative domains. In [20,Theorem 11], a one-to-one correspondence was established between correctable subsystems B of a finite-dimensional channel E : B(S) → B(S) and generalized multiplicative domains MD π (E), where the latter is defined relative to a projection P ∈ P(S), a C*-subalgebra N ⊆ B(P S) and a representation π : N → B(S), to be…”

Nullspaces of Entanglement Breaking Channels and Applications

“…In this section, we build on the nullspace analyses above to derive constructions of algebras privatized by certain entanglement breaking channels. We first review some details of an important operator structure from operator theory [10], which in more recent years has also found a role in quantum information [11,24,29,38,39].…”

“…Building on this, and taking motivation from quantum privacy, we derive a test for mixed unitarity of quantum channels [3,17,22,16,32] based on entanglement breaking channel nullspaces and complementary channel [19,20,21] behaviour. Starting from a connection with channel nullspaces, we also identify conditions that guarantee the existence of private algebras [2,9,5,6,13,26,23,33,14] for certain classes of entanglement breaking channels based on an analysis of multiplicative domains [10,11,24,29,38,39] for the channels.…”

Spectral properties of tensor products of channels

“…In the context of quantum information theory, the multiplicative domain of a channel was studied in connection to the scheme of error correction ( [7], [19]). Specifically, Theorem 11 in [7] states that the multiplicative domain is precisely the algebra over the largest unitarily correctable code (UCC).…”