Abstract
Given a convex function \({\varphi}\) and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by \({\mathcal{H}(A,B)={\rm Tr} \left[ \varphi(A) - \varphi(B) - \varphi'(B)(A-B) \right]}\). Among other things, they prove that the so-defined quantity is monotone if and only if \({\varphi'}\) is operator monotone. The monotonicity is then used to properly define \({\mathcal{H}(A,B)}\) 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 \({\left\lbrace P_n \right\rbrace_{n=1}^{\infty}}\) with \({P_n \to 1}\) strongly, the limit \({\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n)}\) is shown to exist and to be independent of the sequence of projections \({\left\lbrace P_n \right\rbrace_{n=1}^{\infty}}\). The question whether this sequence converges to its "obvious" limit, namely \({{\rm Tr} \left[ \varphi(A)- \varphi(B) - \varphi'(B)(A-B) \right]}\), has been left open. We answer this question in principle affirmatively and show that \({\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n) = {\rm Tr} \left[ \varphi(A) - \varphi(B) - \frac{{\rm d}}{{\rm d} \alpha} \varphi\left( \alpha A + (1-\alpha)B \right)\vert_{\alpha = 0} \right]}\). If the operators A and B are regular enough, that is (A − B), \({\varphi(A)-\varphi(B)}\) and \({\varphi'(B)(A-B)}\) are trace-class, the identity \({{\rm Tr}\Big[ \varphi(A) - \varphi(B) - \frac{{\rm d}}{{\rm d} \alpha} \varphi\left( \alpha A + (1-\alpha)B \right)\vert_{\alpha = 0} \Big] = {\rm Tr} \Big[ \varphi(A)- \varphi(B) - \varphi'(B)(A-B) \Big]}\) holds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Lewin M., Sabin J.: A family of monotone quantum relative entropies. Lett. Math. Phys. 104, 691–705 (2014)
Hainzl C., Lewin M., Seiringer R.: A nonlinear model for relativistic electrons at positive temperature. Rev. Math. Phys. 20, 1283–1307 (2008)
Frank R.L., Hainzl C., Seiringer R., Solovej J.P.: Microscopic derivation of Ginzburg-Landau theory. J. Am. Math. Soc. 25(3), 667–713 (2012)
Bhatia R.: Matrix analysis. Springer-Verlag, New York (1997)
Reed M., Simon B.: Methods of modern mathematical physics. I. Functional analysis. Academic Press, San Diego (1980)
Reed M., Simon B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, San Diego (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
\({\copyright}\)2015 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
Rights and permissions
About this article
Cite this article
Deuchert, A., Hainzl, C. & Seiringer, R. Note on a Family of Monotone Quantum Relative Entropies. Lett Math Phys 105, 1449–1466 (2015). https://doi.org/10.1007/s11005-015-0787-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-015-0787-5