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
Volume: 70
Pages: 5057-5076
No. of pages: 20
ISSN: 0018-9448
Publication date: 07.2024
Publication status: Published
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.",
keywords = "Quantum information science, quantum channel, quantum communication, quantum state",
author = "Niklas Galke and {Van Luijk}, Lauritz and Henrik Wilming",
note = " Publisher Copyright: IEEE",
year = "2024",
month = jul,
doi = "10.48550/arXiv.2304.12989",
language = "English",
volume = "70",
pages = "5057--5076",
journal = "IEEE Transactions on Information Theory",
issn = "0018-9448",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "7",
}