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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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The Connes embedding problem: A guided tour
The Connes embedding problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II 1 _1 factor. The CEP has had interactions with a wide variety of areas of mathematics, including C ∗ \mathrm {C}^* -algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry, to name a few. After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In fact, we outline two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem in C ∗ \mathrm {C}^* -algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.
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- Award ID(s):
- 2054477
- PAR ID:
- 10410828
- Date Published:
- Journal Name:
- Bulletin of the American Mathematical Society
- Volume:
- 59
- Issue:
- 4
- ISSN:
- 0273-0979
- Page Range / eLocation ID:
- 503 to 560
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/bull/1768
