Speaker
Description
In my talk we consider the following topics:
1.Free and C-free probaility and completely positive maps.
2.Free independent projections as a model of Jozef Lukasiewicz
$n$-valued logic , $n>2$ and also model of continuous logic of Lukasiewicz-Tarski.
3.Main Theorem: : If $q$ is real number and $x,y$ are from interval $(0,1)$, then the Tsallis entropy is defined as
$$T_{q} (x,y) = [ x^{1-q} + y^{1-q} -1 ]_{+} ^{^1/(1-q)}.$$
Then we have: If $\mathbf{P}$ and $\mathbf{Q}$ are free independent in some probability space $(\mathbf{A},\text{tr})$ with trace tr state on $\mathbf{A}$, and $\text{tr}(\mathbf{P})=x$, $\text{tr}(yQ) =y$, then $\text{tr}(\mathbf{P}^\mathbf{Q})= T_{0}(x,y)$, if $\mathbf{P}$ and $\mathbf{Q}$ are Boolean independent, then $\text{tr}(\mathbf{P}^\mathbf{Q}) = T_{2} (x,y)$ and relations with Dagum distributions, which are called log-logistic distributions in many statistics models.
If $\mathbf{P}$ and $\mathbf{Q}$ are classical independent then $\text{tr}(\mathbf{P}^\mathbf{Q}) = T_{1}(x,y) =\lim
T_{s} (x,y)$, as s tends to 1.
Here the projection $\mathbf{P}^\mathbf{Q}$ is the smallest projections on the closed linear span of Im($\mathbf{P}$) and Im($\mathbf{Q}$).
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 qunatum channels.
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