percepts}},$$ then we have $$r^{(t)} = \frac{1}{N} \sum_{i=1}^N P^{t} (a_{s_i}^{*} | s_i ).
\label{eqoriggin}$$

There are other important properties in the PS scheme such as edge-glow mechanism $ g(c_i, c_j ) \in [0, 1] $ [@Briegel2; @Alexey1] which refers to a delayed reward $ g^{(t+1)} (c_i, c_j ) \, \lambda^{(t+1)} $ for a non-traversed edge in the right hand of Eq. \[eqtrans\]. In the current study, however, our examples contain only the invasion game for which the agent performs optimally in case we do not have such temporal correlations that are $ g=0 $ [@Briegel3]. Accordingly, the adaptation rule of Eq. \[eqtrans\] is enough for our purpose here.

Projective simulation as a new model for classical and quantum artificial intelligence needs to be expanded in all AI directions as a lot of works have been done so far. Particularly, some classical studies are [@Briegel1; @Briegel2; @Briegel3; @Alexey1] and some quantum works are [@Dunjko1; @Tiersch11; @JensClausen.; @Alexey2; @Dunjko2]. In the current study, we use Dirac notation, which is a useful theoretical method in both classical AI and quantum context, to speak of the influence of a partially observable environment on the efficiency of a given agent, though my results are presented 