Abstract
We study here the elementary properties of the relative entropy \({\mathcal{H}_\varphi(A, B) = {\rm Tr}[\varphi(A) - \varphi(B) - \varphi'(B)(A - B)]}\) for φ a convex function and A, B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ′ is operator monotone. We use this to appropriately define \({\mathcal{H}_\varphi(A, B)}\) in infinite dimension.
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References
Ando T., Petz D.: Gaussian Markov triplets approached by block matrices. Acta Sci. Math.(Szeged) 75, 265–281 (2009)
Audenaert K., Hiai F., Petz D.: Strongly subadditive functions. Acta Mathematica Hungarica 128, 386–394 (2010)
Bach, V., Lieb, E.H., Solovej, J.P.: Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994)
Bhatia R.: Matrix Analysis, vol. 169. Springer, Berlin (1997)
Carlen, E.: Trace inequalities and quantum entropy: an introductory course. In: Sims, R. Ueltschi, D. (eds.) Entropy and the Quantum, vol. 529 of Contemporary Mathematics, American Mathematical Society, 2010, pp. 73–140. Arizona School of Analysis with Applications, March 16–20, 2009, University of Arizona
Davis C.: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc. 8, 42–44 (1957)
Frank R.L., Hainzl C., Seiringer R., Solovej J.P.: Microscopic derivation of Ginzburg-Landau theory. J. Am. Math. Soc. 25, 667–713 (2012)
Hainzl C., Lewin M., Seiringer R.: A nonlinear model for relativistic electrons at positive temperature. Rev. Math. Phys. 20, 1283–1307 (2008)
Hainzl C., Lewin M., Solovej J.P.: The thermodynamic limit of quantum Coulomb systems. Part II. Applications. Adv. Math. 221, 488–546 (2009)
Hansen, F., Pedersen, G.K.: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258, 229–241 (1981/82)
Lewin, M., Sabin, J.: The Hartree equation for infinitely many particles. I. Well-posedness theory. Commun. Math. Phys. (2014) (in press)
Lieb E.H., Ruskai M.B.: A fundamental property of quantum-mechanical entropy. Phys. Rev. Lett. 30, 434–436 (1973)
Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938–1941 (1973) (with an appendix by B. Simon)
Ohya M., Petz D.: Quantum Entropy and its Use. Texts and Monographs in Physics. Springer, Berlin (1993)
Robinson D., Ruelle D.: Mean entropy of states in classical statistical mechanics. Commun. Math. Phys. 5, 288–300 (1967)
Ruelle, D.: Statistical Mechanics. Rigorous Results. World Scientific/Imperial College Press, Singapore/London (1999)
Wehrl A.: General properties of entropy. Rev. Modern Phys. 50, 221–260 (1978)
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Lewin, M., Sabin, J. A Family of Monotone Quantum Relative Entropies. Lett Math Phys 104, 691–705 (2014). https://doi.org/10.1007/s11005-014-0689-y
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DOI: https://doi.org/10.1007/s11005-014-0689-y