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Geometric mean of states and transition amplitudes. (English) Zbl 1172.46044
Summary: The transition amplitude between square roots of states, which is an analogue of Hellinger integral in classical measure theory, is investigated in connection with operator-algebraic representation theory. A variational expression based on geometric mean of positive forms is utilized to obtain an approximation formula for transition amplitudes.

MSC:
46L60 Applications of selfadjoint operator algebras to physics
46L53 Noncommutative probability and statistics
81R15 Operator algebra methods applied to problems in quantum theory
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[1] Alberti, P.M., Uhlmann, A.: On Bures distance and *-algebraic transition probability between inner derived positive linear forms over W*-algebras. Acta Appl. Math. 60, 1–37 (2000) · Zbl 0961.46043
[2] Araki, H.: Some properties of the modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule. Pac. J. Math. 50, 309–354 (1974) · Zbl 0287.46074
[3] Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007) · Zbl 1133.15017
[4] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Heidelberg (1979) · Zbl 0421.46048
[5] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Heidelberg (1979) · Zbl 0421.46048
[6] Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980) · Zbl 0421.47011
[7] Lawson, J.D., Lim, Y.: The geometric mean, matrices, metrics, and more. Am. Math. Mon. 108, 797–812 (2001) · Zbl 1040.15016
[8] Pedersen, G.K.: C*-Algebras and Their Automorphism Groups. Academic Press, London (1979) · Zbl 0416.46043
[9] Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear froms and the purification map. Rep. Math. Phys. 8, 159–170 (1975) · Zbl 0327.46032
[10] Raggio, G.A.: Comparison of Uhlmann’s transition probability with the one induced by the natural cone of von Neumann algebras in standard form. Lett. Math. Phys. 6, 233–236 (1982) · Zbl 0516.46049
[11] Scutaru, H.: A new measure of nonclassical distance. arXiv:quant-ph/9908089
[12] Stratila, S., Zsido, L.: Lectures on von Neumann Algebras. Abacus Press, London (1979)
[13] Takesaki, M.: Theory of Operator Algebras I. Springer, Heidelberg (1979) · Zbl 0436.46043
[14] Takesaki, M.: Theory of Operator Algebras II. Springer, Heidelberg (2001) · Zbl 1021.46043
[15] Uhlmann, A.: The transition probability in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976) · Zbl 0355.46040
[16] Uhlmann, A.: Relative entropy and Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory. Commun. math. Phys. 54, 21–32 (1977) · Zbl 0358.46026
[17] Woronowicz, S.L.: On the purification of factor states. Commun. math. Phys. 28, 221–235 (1972) · Zbl 0244.46075
[18] Yamagami, S.: Algebraic aspects in modular theory. Publ. RIMS 28, 1075–1106 (1992) · Zbl 0809.46075
[19] Yamagami, S.: Modular theory for bimodules. J. Funct. Anal. 125, 327–357 (1994) · Zbl 0816.46043
[20] Yamagami, S.: Geometry of quasifree states of CCR algebras. arXiv:0801.1739 · Zbl 1259.46060
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