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We revisit the connection between von Neumann algebra index and relative entropy. We observe that the Pimsner-Popa index connects to maximal sandwiched p-R\'enyi relative entropy for all p between 1/2 and infinity, including the Umegaki's relative entropy at p=1. Based on that, we introduce a new notation of maximal relative entropy for a inclusion of finite von Neumann algebras. These maximal relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup
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Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings
Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.
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- PAR ID:
- 10459136
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 63
- Issue:
- 12
- ISSN:
- 0022-2488
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0066653
