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arXiv:0811.0947 · Journal of Physics A Mathematical and Theoretical

The multiplicative domain in quantum error correction

Man-Duen Choi, Nathaniel Johnston, David W. Kribs

Published May 28, 2009• 23 citations
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Abstract

We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. In the arbitrary, not necessarily unital case, they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel. As part of the analysis, we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.

DOI
10.1088/1751-8113/42/24/245303

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We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. In the arbitrary, not necessarily unital case, they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel. As part of the analysis, we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.
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.
Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
Reversing the effects of a quantum evolution, for example, as is done in error correction, is an important task for controlling quantum systems in order to produce reliable quantum devices. When the evolution is governed by a completely positive map, there exist reversibility conditions, known as the quantum error correcting code conditions, which are necessary and sufficient conditions for the reversibility of a quantum operation on a subspace, the code space. However, if we suppose that the evolution is not described by a completely positive map, necessary and sufficient conditions are not known. Here we consider evolutions that do not necessarily correspond to a completely positive map. We prove that the completely positive map error correcting code conditions can lead to a code space that is not in the domain of the map, meaning that the output of the map is not positive. A corollary to our theorem provides a class of relevant examples. Finally, we provide a set of sufficient conditions that will enable the use of quantum error correcting code conditions while ensuring positivity.
We define and investigate the notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the coefficients associated with the Knill-Laflamme conditions and the entropy exchange computed with respect to any initial state supported on the code. In general the entropy of a code can be viewed as a measure of how close it is to the minimal entropy case, which is given by unitarily correctable codes (including decoherence-free subspaces), or the maximal entropy case, which from dynamical Choi matrix considerations corresponds to non-degenerate codes. We consider several examples, including a detailed analysis of the case of binary unitary channels, and we discuss an extension of the entropy to operator quantum error correcting subsystem codes.
Noise poses a challenge for any real-world implementation in quantum information science. The theory of quantum error correction deals with this problem via methods to encode and recover quantum information in a way that is resilient against that noise. Unitarily correctable codes are an error correction technique wherein a single unitary recovery operation is applied without the need for an ancilla Hilbert space. Here, we present an optical implementation of a nontrivial unitarily correctable code for a noisy quantum channel with no decoherence-free subspaces or noiseless subsystems. We show that recovery of our initial states is achieved with high fidelity $(\ensuremath{\ge}0.97)$, quantitatively proving the efficacy of this unitarily correctable code.
Quantum replacer codes are codes that can be protected from errors induced by a given set of quantum replacer channels, an important class of quantum channels that includes the erasures of subsets of qubits that arise in quantum error correction. We prove a structure theorem for such codes that synthesizes a variety of special cases with earlier theoretical work in quantum error correction. We present several examples and applications of the theorem, including a mix of new observations and results together with some subclasses of codes revisited from this new perspective.