Quantum information processing under maximal distance measures
Abstract
The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.
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BibTeX
@misc{regula2026quantum,
title = {{Quantum information processing under maximal distance measures}},
author = {Bartosz Regula and Ludovico Lami and Mario Berta and Marco Tomamichel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06842},
eprint = {2610.06842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06842},
}
RIS
TY - UNPB
TI - Quantum information processing under maximal distance measures
AU - Bartosz Regula
AU - Ludovico Lami
AU - Mario Berta
AU - Marco Tomamichel
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06842
UR - https://arxiv.org/abs/2610.06842
AB - The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.
ER -