Abstract
A subalgebraM 0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM 0. The main result says thatM 0 is weakly sufficient for (φ,ω) if and only ifM 0 contains the Radon-Nikodym cocycle [Dφ,Dω] t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Accardi, L., Cecchini, C.: Conditional expectation in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal.45, 245–273 (1982)
Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Inst. Math. Sci.9, 165–209 (1973)
Araki, H.: Recent developments in the theory of operator algebras and their significance in theoretical physics. Symposia Math., XX, 395–424. New York: Academic Press 1976
Araki, H.: Relative entropy of states of von Neumann algebras. I, II. Publ. Res. Inst. Math. Sci.11, 809–833 (1976) and13, 173–192 (1977)
Araki, H., Masuda, T.: Positive connes andL p-spaces for von Neumann algebras. Publ. Res. Inst. Math. Sci.18, 339–411 (1982)
Connes, A.: Une classification des facteurs de type III. Ann. Sci. Ec. Norm. Supér.6, 133–252 (1973)
Csiszár, I.: Information type measure of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung.2, 299–318 (1967)
Davis, C.: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc.8, 42–44 (1957)
Dunford, N., Schwartz, J.T.: Linear operators. Part II. New York: Interscience 1958
Emch, G.G.: Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley 1972
Gudder, S., Merchand, J.P.: Noncommutative probability on von Neumann algebras. J. Math. Phys.13, 799–806 (1972)
Halmos, P.R., Savage, L.J.: Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Stat.20, 225–241 (1949)
Hiai, F., Ohya, M., Tsukada, M.: Sufficiency, KMS condition and relative entropy in von Neumann algebras. Pac. J. Math.96, 99–109 (1981)
Hiai, F., Ohya, M., Tsukada, M.: Sufficiency and relative entropy in *-algebras with applications in quantum systems. Pac. J. Math.107, 117–140 (1983)
Holevo, A.S.: Some estimates for the amount of information transmittable by a quantum communication channel (in Russian). Probl. Peredachi Inf.9, 3–11 (1973)
Holevo, A.S.: Investigations in the general theory of statistical decisions. Am. Math. Soc. Proc. Steklov Inst. Math.124 (1978)
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat.22, 79–86 (1951)
Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math.11, 267–288 (1973)
Lindblad, G.: Entropy, information and quantum measurement. Commun. Math. Phys.33, 305–322 (1973)
Lindblad, G.: Expectation and entropy inequalities for finite quantum systems. Commun. Math. Phys.39, 111–119 (1974)
Petz, D.: A dual in von Neumann algebras with weights. Q. J. Math. Oxf.35, 475–483 (1984)
Petz, D.: Properties of quantum entropy, quantum probability and applications II. Accardi, L., von Waldenfels, W. (eds.). Lecture Notes in Mathematics. Vol. 1136, pp. 428–441. Berlin, Heidelberg, New York: Springer 1985
Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. Res. Inst. Math. Sci.21, 787–800 (1985)
Raggio, G.A.: Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of a von Neumann algebra in standard form. Lett. Math. Phys.6, 233–236 (1982)
Strătilă, S., Zsidó, L.: Lectures on von Neumann algebras. Tunbridge Wells: Abacuss Press 1979
Strătilă, S.: Modular theory of operator algebras. Tunbridge Wells: Abacus Press 1981
Strasser, H.: Mathematical theory of statistics. Berlin: de Gruyter 1985
Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal.9, 306–321 (1972)
Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep.14, 59–85 (1962)
Wolfe, J.C., Emch, G.G.:C*-algebraic formalism for coarse graining. I. General theory. J. Math. Phys.15, 1343–1347 (1974)
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Petz, D. Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun.Math. Phys. 105, 123–131 (1986). https://doi.org/10.1007/BF01212345
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212345