The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences
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abstract
We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the R\'enyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.
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Infinite-dimensional quantum Gibbs samplers for Bose-Hubbard and Coulomb models achieve exponential relative-entropy decay, while a periodic charging model shows a positive spectral gap alone does not imply hypercontractivity.
The authors extend recovery theorems for α-z Rényi divergences from 2-positive maps to merely positive maps between general von Neumann algebras using Jordan algebraic techniques.
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Integral representations of $f$-divergences for general von Neumann algebras
The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.
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Hypercontractivity and entropy decay for infinite-dimensional quantum Gibbs samplers
Infinite-dimensional quantum Gibbs samplers for Bose-Hubbard and Coulomb models achieve exponential relative-entropy decay, while a periodic charging model shows a positive spectral gap alone does not imply hypercontractivity.
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Sufficient positive maps between von Neumann algebras: R\'enyi divergences
The authors extend recovery theorems for α-z Rényi divergences from 2-positive maps to merely positive maps between general von Neumann algebras using Jordan algebraic techniques.
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