Note on a family of monotone quantum relative entropies Journal Article

Author(s): Deuchert, Andreas; Hainzl, Christian; Seiringer, Robert
Article Title: Note on a family of monotone quantum relative entropies
Affiliation IST Austria
Abstract: Given a convex function (Formula presented.) and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by (Formula presented.). Among other things, they prove that the so-defined quantity is monotone if and only if (Formula presented.) is operator monotone. The monotonicity is then used to properly define (Formula presented.) for bounded self-adjoint operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections (Formula presented.) with (Formula presented.) strongly, the limit (Formula presented.) is shown to exist and to be independent of the sequence of projections (Formula presented.). The question whether this sequence converges to its "obvious" limit, namely (Formula presented.), has been left open. We answer this question in principle affirmatively and show that (Formula presented.). If the operators A and B are regular enough, that is (A − B), (Formula presented.) and (Formula presented.) are trace-class, the identity (Formula presented.) holds.
Keywords: operator monotonicity; relative entropy
Journal Title: Letters in Mathematical Physics
Volume: 105
Issue 10
ISSN: 1573-0530
Publisher: Springer  
Date Published: 2015-08-05
Start Page: 1449
End Page: 1466
DOI: 10.1007/s11005-015-0787-5
Open access: yes (repository)
IST Austria Authors
  1. Robert Seiringer
    123 Seiringer
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