Note on a family of monotone quantum relative entropies

A. Deuchert, C. Hainzl, R. Seiringer, Letters in Mathematical Physics 105 (2015) 1449–1466.

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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.
Publishing Year
Date Published
2015-08-05
Journal Title
Letters in Mathematical Physics
Volume
105
Issue
10
Page
1449 - 1466
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Deuchert A, Hainzl C, Seiringer R. Note on a family of monotone quantum relative entropies. Letters in Mathematical Physics. 2015;105(10):1449-1466. doi:10.1007/s11005-015-0787-5
Deuchert, A., Hainzl, C., & Seiringer, R. (2015). Note on a family of monotone quantum relative entropies. Letters in Mathematical Physics, 105(10), 1449–1466. https://doi.org/10.1007/s11005-015-0787-5
Deuchert, Andreas, Christian Hainzl, and Robert Seiringer. “Note on a Family of Monotone Quantum Relative Entropies.” Letters in Mathematical Physics 105, no. 10 (2015): 1449–66. https://doi.org/10.1007/s11005-015-0787-5.
A. Deuchert, C. Hainzl, and R. Seiringer, “Note on a family of monotone quantum relative entropies,” Letters in Mathematical Physics, vol. 105, no. 10, pp. 1449–1466, 2015.
Deuchert A, Hainzl C, Seiringer R. 2015. Note on a family of monotone quantum relative entropies. Letters in Mathematical Physics. 105(10), 1449–1466.
Deuchert, Andreas, et al. “Note on a Family of Monotone Quantum Relative Entropies.” Letters in Mathematical Physics, vol. 105, no. 10, Springer, 2015, pp. 1449–66, doi:10.1007/s11005-015-0787-5.
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