Constructing Entanglers in 2-Players--N-Strategies Quantum Game

In quantum games based on 2-player–$N$-strategies classical games, each player has a quNit (a normalized vector in an $N$-dimensional Hilbert space ${\mathcal{H}}_N$) upon which he applies his strategy (a matrix $U\in$ SU(N)). The players draw their payoffs from a state $|\Psi \ra=J^\dagger U_1 \otimes U_2 J|\Psi_0 \ra \in {\cal H}_N \otimes {\cal H}_N $\mathrm{.}Here$|\Psi_0 \ra$and$J$ (both determined by the game’s referee) are respectively an {\it unentangled} 2-quNit (pure) state and a unitary operator such that $|\Psi_1 \ra \equiv J|\Psi_0 \ra \in {\cal H}_N \otimes {\cal H}_N$ is {\it partially entangled}. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of $\mathrm{\mid }{\mathrm{\Psi }}_1\text{ra}$. Hence, it is practical to design the entangler $J=J(\beta )$ to be dependent on a {\it single} real parameter $\beta$ that controls the degree of entanglement of $\mathrm{\mid }{\mathrm{\Psi }}_1\text{ra}$, such that its von-Neumann entropy ${\mathcal{S}}_N(\beta )$ is continuous and obtains {\it any value} in $[0,\log N]$. Moreover, an efficient control of ${\mathcal{S}}_N(\beta )$ is possible only if $\mathrm{\mid }{\mathrm{\Psi }}_1\text{ra}$ appears in a Schmidt decomposed form. Designing $J(\beta )$ for $N=2$ is quite standard. Extension to $N>2$ is not obvious, and here we suggest an algorithm to achieve it.