Approximate quantum error correction, random codes, and quantum channel capacity

Abstract

We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.

29 pages, final version to appear in Phys. Rev. A, improved lower bound for code entanglement fidelity, simplified proof

References in corpus (12)

• Mixed State Entanglement and Quantum Error Correction
• The capacity of the noisy quantum channel
• A simple formula for the average gate fidelity of a quantum dynamical operation
• Quantum data processing and error correction
• Information transmission through a noisy quantum channel
• The mother of all protocols: Restructuring quantum information's family tree
• Approximate quantum error correction can lead to better codes
• On Quantum Fidelities and Channel Capacities
• A decoupling approach to the quantum capacity
• Information-theoretic approach to quantum error correction and reversible measurement
• On entanglement-assisted classical capacity
• Tema Con Variazioni: Quantum Channel Capacity

Cited by in corpus (26)

• A decoupling approach to the quantum capacity
• Information preserving structures: A general framework for quantum zero-error information
• A simple approach to approximate quantum error correction based on the transpose channel
• Principles of Quantum Communication Theory: A Modern Approach
• Experimentally scalable protocol for identification of correctable codes
• Energy-constrained private and quantum capacities of quantum channels
• Quantum Communication in Rindler Spacetime
• Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding
• On Quantum Capacity of Compound Channels
• Quantum Information Encoding, Protection, and Correction from Trace-Norm Isometries
• Secrecy capacities of compound quantum wiretap channels and applications
• Operator quantum error correction for continuous dynamics
• Quantum Steganography over Noisy Channels: Achievability and Bounds
• The Physics of Quantum Information: Complementarity, Uncertainty, and Entanglement
• Quantum Steganography over Noiseless Channels: Achievability and Bounds
• Simultaneous transmission of classical and quantum information under channel uncertainty and jamming attacks
• Trade-offs on number and phase shift resilience in bosonic quantum codes
• Information transmission with continuous variable quantum erasure channels
• Universal random codes: Capacity regions of the compound quantum multiple-access channel with one classical and one quantum sender
• Approximate quantum error correction, covariance symmetry, and their relation
• Transmission of coherent information at the onset of interactions
• State-adaptive quantum error correction and fault-tolerant quantum computing
• Entropy of a quantum error correction code
• Approximate Quantum Error Correction with 1D Log-Depth Circuits
• Explicit decoders using fixed-point amplitude amplification based on QSVT
• Spectral properties and coding transitions of Haar-random quantum codes