Appears in collection : Combinatorics and Arithmetic for Physics
In my talk, we consider the following topics:
1. Free and C-free probability and completely positive maps.
2. Free independent projections as a model of Jozef Łukasiewicz $n$-valued logic, $n > 2$, and also a model of continuous logic of Łukasiewicz–Tarski.
3. 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.
4. Remarks on the free product of quantum channels.
References:
1. M. Bożejko, Positive definite functions on the free group and the noncommutative Riesz product, Boll. Un. Mat. Ital. (6) 5-A (1986), 13–21.
2. M. Bożejko, Remarks on free projections, Heidelberg Seminar 1999.
3. W. Mlotkowski, Operator-valued version of conditionally free product, Studia Math. 153:13–30, (2002).
4. 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.
5. M. Bożejko, Conditionally free probability, in Signal Proceeding and Hypercomplex Analysis, 139–147, 2019.
