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

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 new and natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational critera, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.

29 pages, 19 examples. Contains complete proofs for all the theorems in arXiv:0705.4282

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• Two Roads to Classicality
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• One-Shot Randomized and Nonrandomized Partial Decoupling
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• Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory
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• Theory of Metastability in Discrete-Time Open Quantum Dynamics
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• Quantum State Merging for Arbitrarily Small-Dimensional Systems
• Distributed Encoding and Decoding of Quantum Information over Networks
• Lindbladians with multiple steady states: theory and applications
• Asymptotic Dynamics of Open Quantum Systems and Modular Theory
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• Genuine entanglement detection via projection map in multipartite systems
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• Protecting coherence from the environment via Stark many-body localization in a Quantum-Dot Simulator
• Exact Model Reduction for Continuous-Time Open Quantum Dynamics
• Reductions in finite-dimensional quantum mechanics: from symmetries to operator algebras and beyond
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• Alternating Projections Methods for Discrete-time Stabilization of Quantum States
• Model Reduction for Quantum Systems: Discrete-time Quantum Walks and Open Markov Dynamics
• Information storage and transmission under Markovian noise
• Unique determination and generation of rank two single-qubit quantum channels
• Structural Characterization And Condition For Measurement Statistics Preservation Of A Unital Quantum Operation
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• Super-activating Quantum Memory with Entanglement
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