# Quantum Conditional Strategies and Automata for Prisoners' Dilemmata under the EWL Scheme
## 6 Quantum Version of Conditional Strategies
### Strategies for the Quantum PD  
Here we establish the strategies for the quantum repeated PD game, and we associate them to inputs of quantum automata. First, the symbolic payoff matrix for the quantum PD game is shown in Table 5, whereas in Table 6, the numerical values for each pair of actions are depicted.  
When multiple stages of a game take place and conditional strategies have to be designed, each player tries to avoid successive losses. To achieve this, Player 1 has to know the action taken by Player 2 in the previous round of the game (and vice versa). This cannot be directly observed by himself since the measurement of the state dissolves any information regarding the choices of the actions of both players. Thus, in our setting, every player announces the unitary operator they used in the previous part, and it is assumed that they are not lying about it.  
As already noted, the pure classical strategies of cooperate and defect are expressed as \(U(0,0)\) and \(U(\pi,0)\), where \(U\) is given by Equation (5). We observe that the cooperate operator  
\[U(0,0)=\left(\begin{array}{cc}1&0\\ 0&1\end{array}\right) \tag{6}\]  
is the identity operator, and the defect operator  
\[U(\pi,0)=\left(\begin{array}{cc}0&1\\ -1&0\end{array}\right) \tag{7}\]  
resembles the bit-flip operator.  
Aside from the two classical actions, we have to introduce two purely quantum actions. The first one is the miracle action [2] that allows the player to always win against the other player's classical strategy.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & & & & & & & \\ \hline \multirow{4}{*}{Player 1} & & \multicolumn{2}{c}{**C**} & \multicolumn{2}{c}{**D**} & \multicolumn{2}{c}{**M**} & \multicolumn{2}{c}{**Q**} \\ \cline{2-6}  & C & \(3,3\) & \(0,5\) & \((0,5)\) or \((1,1)\) & \(1,1\) \\ \cline{2-6}  & D & \(5,0\) & \(1,1\) & \((0,5)\) or \((1,1)\) & \(0,5\) \\ \cline{2-6}  & M & \((5,0)\) or \((1,1)\) & \((5,0)\) or \((1,1)\) & All & \((3,3)\) or \((0,5)\) \\ \cline{2-6}  & Q & \(1,1\) & \(5,0\) & \((3,3)\) or \((5,0)\) & \(3,3\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: The numerical values for the quantum PD game.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & & & & & & \\ \hline \multirow{4}{*}{Player 1} & & \multicolumn{2}{c}{**C**} & \multicolumn{2}{c}{**D**} & \multicolumn{2}{c}{**M**} & \multicolumn{2}{c}{**Q**} \\ \cline{2-6}  & C & \(r,r\) & \(s,t\) & \((s,t)\) or \((p,p)\) & \(p,p\) \\ \cline{2-6}  & D & \(t,s\) & \(p,p\) & \((s,t)\) or \((p,p)\) & \(s,t\) \\ \cline{2-6}  & M & \((t,s)\) or \((p,p)\) & \((t,s)\) or \((p,p)\) & All & \((r,r)\) or \((s,t)\) \\ \cline{2-6}  & Q & \(p,p\) & \(t,s\) & \((r,r)\) or \((t,s)\) & \(r,r\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The payoff matrix for the quantum PD game, where players can choose to apply four different actions. C stands for cooperate, D for defect, M for the miracle move, and Q for the Q move.  
The other action is the Q move, which enables the existence of a new Nash equilibrium point that is also a Pareto optimal. We associate the player's strategies with the following unitary matrices:  
\[\begin{array}{c}C=\begin{pmatrix}1&0\\ 0&1\end{pmatrix},\qquad D=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\\ \\ M=\frac{1}{\sqrt{2}}\begin{pmatrix}i&1\\ -1&-i\end{pmatrix},\quad Q=\begin{pmatrix}i&0\\ 0&-i\end{pmatrix}.\end{array} \tag{8}\]