Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1004.5112 · Proceedings of the American Mathematical Society

Generalized multiplicative domains and quantum error correction

Nathaniel Johnston, David W. Kribs

Published Oct 27, 2010• 12 citations
arXivHTMLPDF

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.

DOI
10.1090/s0002-9939-2010-10556-7

Related papers

The multiplicative domain in quantum error correction

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

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.

arXiv:0811.0947•Journal of Physics A Mathematical and Theoretical•May 28, 2009•23 citations

Multiplicative properties of quantum channels

Mizanur Rahaman

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.

arXiv:1701.06205•Journal of Physics A Mathematical and Theoretical•Jun 23, 2017•20 citations

Completely positive maps and passive quantum error correction

Aron Pasieka

We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. The Lindblad master equation is presented as an example of the dynamics of a system thusly described, in the case of a Markovian system. We review noiseless subsystems and decoherence-free subspaces. Building on the generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems, given a specific, not necessarily unital, channel. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to find unitarily correctable subsystems, in the case of a unital noise model.

•Jan 1, 2007•0 citations

Operator quantum error correction

David W. Kribs, Raymond Laflamme, David Poulin +1 more

This paper is an expanded and more detailed version of the work \cite{KLP04} 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''.

Quantum Information and Computation•Jul 1, 2006•73 citations

Sufficient conditions and constraints for reversing general quantum errors

Alvin Gonzales, Daniel Dilley, Mark Byrd

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.

arXiv:2007.11083•Physical Review A•Dec 21, 2020•3 citations

Operator quantum error correction

David W. Kribs, Raymond Laflamme, David Poulin +1 more

This paper is an expanded and more detailed version of our recent work 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''.

arXiv:quant-ph/0504189•Apr 26, 2005•2 citations
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.
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.
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.
We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. The Lindblad master equation is presented as an example of the dynamics of a system thusly described, in the case of a Markovian system. We review noiseless subsystems and decoherence-free subspaces. Building on the generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems, given a specific, not necessarily unital, channel. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to find unitarily correctable subsystems, in the case of a unital noise model.
This paper is an expanded and more detailed version of the work \cite{KLP04} 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''.
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.
This paper is an expanded and more detailed version of our recent work 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''.