Details
| Original language | English |
|---|---|
| Article number | 10472942 |
| Pages (from-to) | 5057-5076 |
| Number of pages | 20 |
| Journal | IEEE Transactions on Information Theory |
| Volume | 70 |
| Issue number | 7 |
| Early online date | 14 Mar 2024 |
| Publication status | Published - Jul 2024 |
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.
Keywords
- Quantum information science, quantum channel, quantum communication, quantum state
ASJC Scopus subject areas
- Computer Science(all)
- Information Systems
- Computer Science(all)
- Computer Science Applications
- Social Sciences(all)
- Library and Information Sciences
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In: IEEE Transactions on Information Theory, Vol. 70, No. 7, 10472942, 07.2024, p. 5057-5076.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Sufficiency of Rényi divergences
AU - Galke, Niklas
AU - Van Luijk, Lauritz
AU - Wilming, Henrik
N1 - Publisher Copyright: IEEE
PY - 2024/7
Y1 - 2024/7
N2 - 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.
AB - 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.
KW - Quantum information science
KW - quantum channel
KW - quantum communication
KW - quantum state
UR - http://www.scopus.com/inward/record.url?scp=85187995788&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2304.12989
DO - 10.48550/arXiv.2304.12989
M3 - Article
AN - SCOPUS:85187995788
VL - 70
SP - 5057
EP - 5076
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
SN - 0018-9448
IS - 7
M1 - 10472942
ER -