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:1006.1358 · Physical Review A

Information-preserving structures: A general framework for quantum zero-error information

Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola

Published Dec 7, 2010• 112 citations
arXivHTMLPDF

Abstract

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., ``noiseless,'' ``unitarily correctible,'' etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.

DOI
10.1103/physreva.82.062306

Related papers

Characterizing the Structure of Preserved Information in Quantum Processes

Robin Blume-Kohout, Hui Khoon Ng, David Poulin +1 more

We introduce a general operational characterization of information-preserving structures-encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes-by demonstrating that they are isometric to fixed points of unital quantum processes. Using this, we show that every information-preserving structure is a matrix algebra. We further establish a structure theorem for the fixed states and observables of an arbitrary process, which unifies the Schrödinger and Heisenberg pictures, places restrictions on physically allowed kinds of information, and provides an efficient algorithm for finding all noiseless and unitarily noiseless subsystems of the process.

arXiv:0705.4282•Physical Review Letters•Jan 22, 2008•97 citations

Stabilizing quantum information

Paolo Zanardi A

The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.

arXiv:quant-ph/9910016•Physical Review A•Dec 5, 2000•207 citations

Theory of Quantum Error Correction for General Noise

Emanuel H. Knill, Raymond Laflamme, Lorenza Viola

A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.

arXiv:quant-ph/9604034•Physical Review Letters•Mar 13, 2000•866 citations

The Theory of Records: A General Principle for Information Protection in Quantum Dynamics

Mustafa Boudjerada

This work formulates and tests a general principle governing when physical records remain accessible under quantum dynamics. Across previous models involving gauge theory, thermalization, closed universes, and scrambling systems, a common structure emerges: some distinctions survive only when the dynamics preserve an observer-accessible information structure. The paper introduces a rigorous definition of records using restricted distinguishability between histories: E_A = − ln F(ρ₀|A, ρ₁|A) and defines protection as the persistence of this distinguishability under an allowed class of quantum channels. The proposed principle is: A record is protected if and only if an observer-accessible Heisenberg fixed-point (peripheral) operator of the allowed channel class distinguishes the histories. The principle is tested against several systems: 1. Covariant quantum channels, where explicit fidelity lower bounds are verified. 2. Many-body localization, showing that protection can arise from emergent quasi-local conserved structures rather than fundamental symmetries. 3. Noiseless subsystems, demonstrating that protected records can exist through operator-algebraic commutants rather than conserved densities. 4. Blind searches over random semigroups, where every protected case contains the predicted fixed-point witness. The work connects quantum records with existing theories of information-preserving structures, superselection, asymmetry, quantum error correction, and emergent conservation laws. The result is a general operator-algebraic framework for understanding why some information survives dynamics while other information becomes inaccessible.

•Aug 26, 2026•0 citations

Theory of quantum error-correcting codes

Emanuel H. Knill, Raymond Laflamme

Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.

Physical Review A•Feb 1, 1997•1,593 citations

Phase Retrieval and Fidelity Preserving Quantum Channels

Liu, Kai

This thesis investigates the preservation of fundamental quantum properties under noisy quantum processes. Motivated by the challenges of maintaining coherence and information integrity in realistic quantum systems, we study three distinct structural properties: phase retrievability, orbit injectivity, and fidelity preservation. Each property reflects a unique facet of how quantum information can be robustly encoded, distinguished, or compared in the presence of environmental interactions. In the first part of the thesis, we analyze phase retrievable quantum channels which allow recovery of a quantum state up to a global phase. We provide structural characterizations using the Kraus representation and the joint spectrum of the Kraus operators. We then extend this study to orbit-injective channels, which distinguish quantum states up to more general group actions. This generalization captures symmetry-induced ambiguities and deepens the understanding of information distinguishability in structured quantum systems. The second part focuses on fidelity, a widely used measure of quantum similarity. We explore conditions under which mixed unitary quantum channels preserve fidelity between quantum states. These investigations reveal how fidelity preservation can constrain the structure of a quantum channel, providing insight into error tolerance and stability in quantum communication. Together, these results contribute to a unified perspective on how different aspects of quantum information-state reconstruction, distinguishability, and similarity-can be maintained under noise, offering both theoretical insight and tools for practical quantum technologies.

Journal of International Crisis and Risk Communication Research•Aug 15, 2025•0 citations
Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., noiseless,'' unitarily correctible,'' etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.
We introduce a general operational characterization of information-preserving structures-encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes-by demonstrating that they are isometric to fixed points of unital quantum processes. Using this, we show that every information-preserving structure is a matrix algebra. We further establish a structure theorem for the fixed states and observables of an arbitrary process, which unifies the Schrödinger and Heisenberg pictures, places restrictions on physically allowed kinds of information, and provides an efficient algorithm for finding all noiseless and unitarily noiseless subsystems of the process.
The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.
A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.
This work formulates and tests a general principle governing when physical records remain accessible under quantum dynamics. Across previous models involving gauge theory, thermalization, closed universes, and scrambling systems, a common structure emerges: some distinctions survive only when the dynamics preserve an observer-accessible information structure. The paper introduces a rigorous definition of records using restricted distinguishability between histories: E_A = − ln F(ρ₀|A, ρ₁|A) and defines protection as the persistence of this distinguishability under an allowed class of quantum channels. The proposed principle is: A record is protected if and only if an observer-accessible Heisenberg fixed-point (peripheral) operator of the allowed channel class distinguishes the histories. The principle is tested against several systems: 1. Covariant quantum channels, where explicit fidelity lower bounds are verified. 2. Many-body localization, showing that protection can arise from emergent quasi-local conserved structures rather than fundamental symmetries. 3. Noiseless subsystems, demonstrating that protected records can exist through operator-algebraic commutants rather than conserved densities. 4. Blind searches over random semigroups, where every protected case contains the predicted fixed-point witness. The work connects quantum records with existing theories of information-preserving structures, superselection, asymmetry, quantum error correction, and emergent conservation laws. The result is a general operator-algebraic framework for understanding why some information survives dynamics while other information becomes inaccessible.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
This thesis investigates the preservation of fundamental quantum properties under noisy quantum processes. Motivated by the challenges of maintaining coherence and information integrity in realistic quantum systems, we study three distinct structural properties: phase retrievability, orbit injectivity, and fidelity preservation. Each property reflects a unique facet of how quantum information can be robustly encoded, distinguished, or compared in the presence of environmental interactions. In the first part of the thesis, we analyze phase retrievable quantum channels which allow recovery of a quantum state up to a global phase. We provide structural characterizations using the Kraus representation and the joint spectrum of the Kraus operators. We then extend this study to orbit-injective channels, which distinguish quantum states up to more general group actions. This generalization captures symmetry-induced ambiguities and deepens the understanding of information distinguishability in structured quantum systems. The second part focuses on fidelity, a widely used measure of quantum similarity. We explore conditions under which mixed unitary quantum channels preserve fidelity between quantum states. These investigations reveal how fidelity preservation can constrain the structure of a quantum channel, providing insight into error tolerance and stability in quantum communication. Together, these results contribute to a unified perspective on how different aspects of quantum information-state reconstruction, distinguishability, and similarity-can be maintained under noise, offering both theoretical insight and tools for practical quantum technologies.