Sufficiency of Rényi divergences

authored by: Niklas Galke, Lauritz Van Luijk, Henrik Wilming
Abstract: A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) Rényi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal.
Organisation(s): Institute of Theoretical Physics
QuantumFrontiers
External Organisation(s): Autonomous University of Barcelona (UAB)
Type: Article
Journal: IEEE Transactions on Information Theory
ISSN: 0018-9448
Publication date: 14.03.2024
Publication status: E-pub ahead of print
Peer reviewed: Yes
ASJC Scopus subject areas: Information Systems, Computer Science Applications, Library and Information Sciences
Electronic version(s): https://doi.org/10.48550/arXiv.2304.12989 (Access: Open)
https://doi.org/10.1109/TIT.2024.3376395 (Access: Closed)

BibTeX

@article{7f69b7e06bce4151ab7adebbc32aed55,
title = "Sufficiency of R{\'e}nyi divergences",
abstract = "A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) R{\'e}nyi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal.",
author = "Niklas Galke and {Van Luijk}, Lauritz and Henrik Wilming",
note = "Acknowledgements We would like to thank Bjarne Bergh, Milan Mosonyi, Robert Salzmann, Joseph Schindler, Alexander Stottmeister, David Sutter, Marco Tomamichel, Reinhard F. Werner and Andreas Winter for interesting and helpful discussions. We also want to thank the two anonymous referees for correc- tions and several valuable suggestions that helped us improve the manuscript. N.G. acknowledges financial support by the MICIIN with funding from European Union NextGenera- tionEU (PRTR-C17.I1) and by the Generalitat de Catalunya. H.W. acknowledges support by the DFG through SFB 1227 (DQ-mat), Quantum Valley Lower Saxony, and funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanys Excellence Strategy EXC-2123 QuantumFrontiers 390837967. ",
year = "2024",
month = mar,
day = "14",
doi = "10.48550/arXiv.2304.12989",
language = "English",
journal = "IEEE Transactions on Information Theory",
issn = "0018-9448",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
}