Quantum information processing under maximal distance measures

arXiv (Cornell University)

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.

Authors 4

1. Bartosz Regula
   Affiliation as printed

   Mathematical Quantum Information RIKEN Hakubi Research Team , RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC) , Wako , Japan

2. Ludovico Lami

   Scuola Normale Superiore

   Affiliation as printed

   Scuola Normale Superiore , Piazza dei Cavalieri 7 , 56126 Pisa , Italy

3. Mario Berta Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

4. Marco Tomamichel

   National University of Singapore · Centre for Quantum Technologies

   Affiliation as printed

   Centre for Quantum Technologies , National University of Singapore , Singapore

   Department of Electrical and Computer Engineering , National University of Singapore , Singapore