Higher Rank Matricial Ranges and Hybrid Quantum Error Correction

Abstract

We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty, and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.

arXiv comment11 pages

References in corpus (19)

• Quantum Computing in the NISQ era and beyond
• Mixed State Entanglement and Quantum Error Correction
• Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• A Theory of Quantum Error-Correcting Codes
• Bulk Locality and Quantum Error Correction in AdS/CFT
• The Ryu-Takayanagi Formula from Quantum Error Correction
• A Unified and Generalized Approach to Quantum Error Correction
• Entanglement-assisted communication of classical and quantum information
• Generalization of Quantum Error Correction via the Heisenberg Picture
• Quantum Error Correcting Codes From The Compression Formalism
• Quantum Error Correction of Observables
• A Holographic Entanglement Entropy Conjecture for General Spacetimes
• Code properties from holographic geometries
• Classical Enhancement of Quantum Error-Correcting Codes
• Higher rank numerical ranges and low rank perturbations of quantum channels
• Simultaneous classical-quantum capacities of quantum multiple access channels
• Efficient Quantum Error Correction for Fully Correlated Noise
• Error correction schemes for fully correlated quantum channels protecting both quantum and classical information