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 ${\cal H}_N$) upon which he applies his strategy (a matrix $U \in$ SU(N)). The players draw their payoffs from a state $|\Psi \ra=J^\dagger U_1 \otimes U_2 J|\Psi_0 \ra \in {\cal H}_N \otimes {\cal H}_N$. 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 $|\Psi_1 \ra$. 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 $|\Psi_1 \ra$, such that its von-Neumann entropy ${\cal S}_N(\beta)$ is continuous and obtains {\it any value} in $[0, \log N]$. Moreover, an efficient control of ${\cal S}_N(\beta)$ is possible only if $|\Psi_1 \ra$ 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.