Orateur
Description
In my talk, we consider the following topics:
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Free and C-free probability and completely positive maps.
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Free independent projections as a model of Jozef Łukasiewicz $n$-valued logic, $n > 2$, and also a model of continuous logic of Łukasiewicz–Tarski.
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Main Theorem: If $q$ is a real number and $x, y$ are from interval $(0, 1)$, then the Tsallis entropy is defined as
$$ T_q (x, y) = (x_1-q + y_1-q-1)^{\frac{1}{1-q}} $$ Then we have: If $P$ and $Q$ are free independent in some probability space $(A, tr)$ with trace $tr$ state on $A$, and $tr(P) = x$, $tr(yQ) = y$, then $tr(P^Q) = T_0(x, y)$. If $P$ and $Q$ are Boolean independent, then $tr(P Q) = T_2(x, y)$ and relations with Dagum distributions, which are called log-logistic distributions in many statistics models. If $P$ and $Q$ are classically independent then $tr(P^Q) = T_1(x, y) = lim_{q\to 1}T_S (x, y)$, as t tends to 1. Here the projection $P^Q$ is the smallest projection on the closed linear span of $Im(P)$ and $Im(Q)$. The generalizations of cases of Tsallis entropy $T_q$, for $q$ in $(0, 1)$, we will use conditionally free independent projections. -
Remarks on the free product of quantum channels.
References:
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M. Bożejko, Positive definite functions on the free group and the noncommutative Riesz product, Boll. Un. Mat. Ital. (6) 5-A (1986), 13–21.
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M. Bożejko, Remarks on free projections, Heidelberg Seminar 1999.
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W. Mlotkowski, Operator-valued version of conditionally free product, Studia Math. 153:13–30, (2002).
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M. Bożejko, Projections in free and Boolean probability with applications to J. Lukasiewicz logic, Conference on Quantum Statistics and Related Topics, Lodz, 10 pp., 2018.
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M. Bożejko, Conditionally free probability, in Signal Proceeding and Hypercomplex Analysis, 139–147, 2019.
