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McMillan type convergence for quantum Gibbs states

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References

  1. H. Araki, Gibbs states of a one dimensional quantum lattice. Comm. Math. Phys.14, 120–157 (1969).

    Google Scholar 

  2. H.Araki, Positive cone, Radon-Nikodym theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of the Tomita-Takesaki theory. In: Proc. Internat. School of Physics “Enrico Fermi”, Course LX, Varenna, 64–100, Amsterdam 1976.

  3. P.Billingsley, Ergodic Theory and Information. New York-London-Sidney 1965.

  4. O.Bratteli and D. W.Robinson, Operator Algebras and Quantum Statistical Mechanics II. Berlin-Heidelberg-New York 1981.

  5. J. P. Conze andN. Dang-Ngoc, Ergodic theorems for noncommutative dynamical systems. Invent. Math.46, 1–15 (1978).

    Google Scholar 

  6. G. G. Emch, Positivity of theK-entropy on non-abelianK-flows. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete29, 241–252 (1974).

    Google Scholar 

  7. H. Föllmer, On entropy and information gain in random fields. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete26, 207–217 (1973).

    Google Scholar 

  8. F. Hiai andD. Petz, Entropy densities for Gibbs states of quantum spin systems. Rev. Math. Phys.5, 693–712 (1994).

    Google Scholar 

  9. R.Jajte, Strong Limit Theorems in Non-Commutative Probability. LNM1110, Berlin-Heidelberg-New York 1985.

  10. U.Krengel, Ergodic Theorems. Berlin-New York 1985.

  11. E. C. Lance, Ergodic theorem for convex sets and operator algebras. Invent. Math.37, 201–214 (1976).

    Google Scholar 

  12. M.Ohya, Entropy operators and McMillan type convergence theorems in a noncommutative dynamical system. In: Probability Theory and Mathematical Statistics, S. Watanabe, Y. V. Prokhorov, eds., LNM1299, 384–390, Berlin-Heidelberg-New York 1988.

  13. M.Ohya and D.Petz, Quantum Entropy and Its Use. Berlin-Heidelberg-New York 1993.

  14. M. Ohya, M. Tsukada andH. Umegaki, A formulation of noncommutative McMillan theorem. Proc. Japan Acad. Ser. A Math. Sci.63, 50–53 (1987).

    Google Scholar 

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Author notes

  1. Dénes Petz

    Present address: Department of Mathematics Faculty of Chemical Engineering, Technical University Budapest, Stoczek u. 2, H-1521, Budapest, Hungary

  2. Masanori Ohya

    Present address: Department of Information Sciences, Science University of Tokyo, Chiba 278, Noda City, Japan

  3. Fumio Hiai

    Present address: Department of Mathematics, Ibaraki University, Ibaraki 310, Mito, Japan

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    1. Fumio Hiai
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    2. Masanori Ohya
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    3. Dénes Petz
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    Additional information

    Partially supported by the Hungarian National Foundation of Scientific Research, grant No. 1900.

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    Hiai, F., Ohya, M. & Petz, D. McMillan type convergence for quantum Gibbs states. Arch. Math 64, 154–158 (1995). https://doi.org/10.1007/BF01196636

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