2014/02/09 by Y. Avishai, Avishai, Y. · 1 citation
Physics and Astronomy · Computer Science · #Quantum Mechanics and Applications #Quantum Information and Cryptography #Quantum Computing Algorithms and Architecture
paper · pdf · doi:10.48550/arxiv.1402.1982
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 HN) upon which he applies his strategy (a matrix U ∈ SU(N)). The players draw their payoffs from a state |Ψ\ra=J^† U1 ⊗ U2 J|Ψ0 \ra ∈ \cal HN ⊗ \cal HN . Here |Ψ0 \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 |Ψ1 \ra ≡ J|Ψ0 \ra ∈ \cal HN ⊗ \cal HN is \it partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of |Ψ1 \ra. Hence, it is practical to design the entangler J=J(β) to be dependent on a \it single real parameter β that controls the degree of entanglement of |Ψ1 \ra, such that its von-Neumann entropy \cal SN(β) is continuous and obtains \it any value in [0, log N]. Moreover, an efficient control of \cal SN(β) is possible only if |Ψ1 \ra appears in a Schmidt decomposed form. 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.