Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes

We show that entirely quantum Shannon theoretic methods, based on von Neumann entropies and their properties, can be used to derive Singleton bounds on the performance of entanglement-assisted hybrid classical-quantum (EACQ) error correcting codes. Concretely, we show that the triple-rate region of...

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Bibliographic Details
Published in:IEEE transactions on information theory 2023-09, Vol.69 (9), p.1-1
Main Authors: Mamindlapally, Manideep, Winter, Andreas
Format: Article
Language:English
Subjects:
Codes
Decoding
entanglement-assisted code
Error correcting codes
Error correction
Error correction codes
Hilbert space
hybrid code
Noise measurement
Quantum entanglement
Quantum error correcting code
Quantum system
Qubit
Qubits (quantum computing)
Singleton bound
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Summary:We show that entirely quantum Shannon theoretic methods, based on von Neumann entropies and their properties, can be used to derive Singleton bounds on the performance of entanglement-assisted hybrid classical-quantum (EACQ) error correcting codes. Concretely, we show that the triple-rate region of qubits, cbits and ebits of possible EACQ codes over arbitrary alphabet sizes is contained in the quantum Shannon theoretic rate region of an associated memoryless erasure channel, which turns out to be a polytope. We show that a large part of this region is attainable by certain EACQ codes, whenever the local alphabet size (i.e. Hilbert space dimension) is large enough, in keeping with known facts about classical and quantum maximum distance separable (MDS) codes: in particular, all of its extreme points and all but one of its extremal lines. The attainability of the remaining one extremal line segment is left as an open question.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3277808