# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 2 The Eisert-Wilkens-Lewenstein (EWL) Scheme  
In this section, we briefly review the quantum game scheme based on the idea of [PERSON], [PERSON] and [PERSON] [4] in its generalized form similar to [14]. The EWLscheme concerns \(2\times 2\) bimatrix games. A \(2\times 2\) bimatrix game is a two-player game with two strategies for each player. It can be described by a matrix  
\[(A,B)= \begin{pmatrix}(a_{00},b_{00})&(a_{01},b_{01})\\ (a_{10},b_{10})&(a_{11},b_{11})\end{pmatrix} . \tag{1}\]  
The first player's strategies are identified with the rows and the second player's with the columns. If the first player chooses row \(k\) and the second player chooses column \(l\), their resulting payoffs are \(a_{kl}\) and \(b_{kl}\), respectively. We consider the Eisert-Wilkens-Lewenstein approach to the game (1) defined by a triple  
\[(N,(D_{p})_{p\in N},(\Pi_{p})_{p\in N}), \tag{2}\]  
where  
* \(N=\{a,b\}\) is the set of players;
* \(D_{p}=\big{\{}(\theta_{p},\phi_{p}):\theta_{p}\in[0,\pi]\) and \(\phi_{p}\in[0,2\pi)\big{\}}\); \(p\in\{a,b\}\), are sets of players' strategies that define 2-parameter unitary operators \[U(\theta_{p},\phi_{p})= \begin{pmatrix}e^{-i\phi_{p}/2}\cos\frac{\theta_{p}}{2}&- e^{-i\phi_{p}/2}\sin\frac{\theta_{p}}{2}\\ e^{i\phi_{p}/2}\sin\frac{\theta_{p}}{2}&e^{i\phi_{p}/2}\cos\frac{\theta_{p}}{2} \end{pmatrix} \in\mathsf{SU}(2);\] (3)
* \(\Pi_{p}\colon D_{a}\times D_{b}\to\mathbb{R}\) are the players' quantum payoff functions given by \[\Pi_{a}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\sum_{k,l=0}^{1}P_{k,l}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) a_{kl},\] (4) \[\Pi_{b}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\sum_{k,l=0}^{1}P_{k,l}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) b_{kl},\] (5) where \(a_{kl}\) and \(b_{kl}\), \(k,l\in\{0,1\}\), are the payoffs of the initial game (1) and \[P_{0,0}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\big{(}\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}\cos \Bigl{(}\frac{\theta_{a}+\phi_{b}}{2}\Bigr{)}+\sin\frac{\theta_{a}}{2}\sin \frac{\theta_{b}}{2}\sin\Bigl{(}\frac{\theta_{a}+\phi_{b}}{2}\Bigr{)}\Bigr{)} ^{2},\] (6) \[P_{0,1}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\big{(}\cos\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2}\cos \Bigl{(}\frac{\theta_{a}-\phi_{b}}{2}\Bigr{)}-\sin\frac{\theta_{a}}{2}\cos \frac{\theta_{b}}{2}\sin\Bigl{(}\frac{\theta_{a}-\phi_{b}}{2}\Bigr{)}\Bigr{)} ^{2},\] \[P_{1,0}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\big{(}\cos\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2}\sin \Bigl{(}\frac{\theta_{a}-\phi_{b}}{2}\Bigr{)}+\sin\frac{\theta_{a}}{2}\cos \frac{\theta_{a}}{2}\cos\Bigl{(}\frac{\theta_{a}-\phi_{b}}{2}\Bigr{)}\Bigr{)} ^{2},\] \[P_{1,1}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}) =\big{(}\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}\sin \Bigl{(}\frac{\theta_{a}+\phi_{b}}{2}\Bigr{)}-\sin\frac{\theta_{a}}{2}\sin \frac{\theta_{b}}{2}\cos\Bigl{(}\frac{\theta_{a}+\phi_{b}}{2}\Bigr{)}\Bigr{)} ^{2}.\]  
The coefficients \(P_{k,l}(\theta_{a},\phi_{a},\theta_{b},\phi_{b})\in[0,1]\) are probabilities depending on the players' strategies (3) and can be derived from [14]  
\[P_{k,l}(\theta_{a},\phi_{a},\theta_{b},\phi_{b})=\Big{|}\langle kl|J\dagger^{ \mathsf{t}}(U(\theta_{a},\phi_{a})\ \otimes\ U(\theta_{b},\phi_{b}))J\ |00\rangle\Big{|}^{2}, \tag{7}\]  
where \(J=\frac{1}{\sqrt{2}}I\otimes I+\frac{i}{\sqrt{2}}\sigma_{x}\otimes\sigma_{x}\) is an entangling operator. As all transformations in (7) are unitary, the coefficients must be normalized  
\[\sum_{k,l=0}^{1}P_{k,l}(\theta_{a},\phi_{a},\theta_{b},\phi_{b})=1,\quad\theta _{a},\theta_{b}\in[0,\pi]\text{ and }\phi_{a},\phi_{a}\in[0,2\pi). \tag{8}\]  
Quantization (2) in parameterization (3) has an essential feature: that it is invariant under isomorphic transformations of the input classical game [20]. If the two classical games are strongly isomorphic, e.g., they differ in the order of players or the order of their strategies, then the corresponding quantum extensions should also be strongly isomorphic.