Abstract
In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability theory (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6, 1]) has an analogue for the free entropy χ(X) (Theorem 2.1).
The free entropy, consistent with Boltzmann's formulaS=klogW, was defined via volumes of matricial microstates. Proving the free entropy power inequality naturally becomes a geometric question.
Restricting the Minkowski sum of two sets means to specify the set of pairs of points which will be added. The relevant inequality, which holds when the set of addable points is sufficiently large, differs from the Brunn-Minkowski inequality by having the exponent 1/n replaced by 2/n. Its proof uses the rearrangement inequality of Brascamp-Lieb-Lüttinger ([2]). Besides the free entropy power inequality, note that the inequality for restricted Minkowski sums may also underlie the classical Shannon entropy power inequality (see 3.2 below).
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References
Blachman, N.M.: The convolution inequality for entropy powers. IEEE Trans. Inform. Theory2, 267–327 (1965)
Brascamp, H.J., Lieb, E.H., Luttinger, J.M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal.17, 227–237 (1974)
Carlen, E.A.: Superadditivity of Fisher's information and logarithmic Sobolev inequalities. J. Funct. Anal.101, 194–211 (1991)
Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge: Cambridge Univ. Press, 1989
Schneider, R.: Convex bodies: The Brunn-Minkowski theory. Encyclopedia of Mathematics and Its Applications44, Cambridge: Cambridge Univ. Press, 1993
Shannon, C.E., Weaver, W.: The Mathematical Theory of Communications. University of Illinois Press, 1963
Stam A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control2, 101–112 (1959)
Voiculescu, D.: The analogues of entropy and of Fisher's information measure in free probability theory, I. Commun. Math. Phys.155, 71–92 (1993); ibidem Voiculescu, D.: The analogues of entropy and of Fisher's information measure in free probability theory, II. Invent. Math.118, 411–440 (1994);ibidem Voiculescu, D.: The analogues of entropy and of Fisher's information measure in free probability theory, III, IHES, preprint, May 1995 (to appear in GAFA)
Voiculescu, D., Dykema, D., Nica, A.: Free Random Variables. CRM Monograph Series, Vol. 1, Providence, RI: American Mathematical Society, 1992
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Communicated by A. Connes
Research supported in part by grants from the National Science Foundation.
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Szarek, S.J., Voiculescu, D. Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality. Commun.Math. Phys. 178, 563–570 (1996). https://doi.org/10.1007/BF02108815
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DOI: https://doi.org/10.1007/BF02108815