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 nor-malized vector in an N-dimensional Hilbert space N ) upon which he applies his strategy (a ma-trix ()U NSU ∈). The players draw their payoffs from a state † 1 2 0 N NJ U U JΨ = ⊗ Ψ ∈ ⊗ . Here 0Ψ and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that 1 0 N NJΨ ≡ Ψ ∈ ⊗ is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of 1Ψ. Hence, it is practical to design the entangler ()J J = β to be de-pendent on a single real parameter β that controls the degree of entanglement of 1Ψ, such that its von-Neumann entropy ()NS β is continuous and obtains any value in []N0, log. Designing ()J β for N 2 = is quite standard. Extension to N 2> is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols