} \, | b_j \rangle ,
\label{eqktswmtt}$$ $$(S+B)_j = \, | (S+B)_{j} \rangle \,\, \langle (S+B)_{j} | .
\label{thefifi}$$ Then, one can define the efficiency of Eq.\[kabli\] as the real part of $ r^{(t)} $ that is $$r^{(t)}= \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*} | (S+B)_j | s_i \rangle$$ $$\begin{aligned}
efficiency \equiv Real \, [ r^{(t)} ] = \frac{\alpha }{N} \sum_{i=1}^N \, \langle a_{s_i}^{*} | {s_i} \rangle \, + \, \frac{1 - \alpha }{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*} | b_j \rangle \, \langle b_j | s_i \rangle .
\label{jankan}\end{aligned}$$ Equation \[eqktswmtt\] is well-defined. Because the operators as $ | b_j \rangle \langle s_j | $ and $ | s_j \rangle \langle b_j | $ for $ b_j \neq s_j  $ cause seemingly nonsensical probabilities in the current study. The former would bring probabilities from which when a direct world state $ s_j $ was seen, the action takes place for an indirect $ b_j $, and the effect is just vice versa for the latter.

Suppose that we have asymptotic probabilities as a function of forgetting factor, $${\langle a_k | s_l \rangle}_{max} = {\langle a_k | b_l \rangle}_{max} = 
\begin{cases}
p ( \gamma_{pa} ) , \quad k = l 