Quantum prisoners' dilemma under enhanced interrogation

George Siopsis, Radhakrishnan Balu, Neal Solmeyer
2018 Quantum Information Processing  
In the quantum version of prisoners' dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, there exist Nash equilibria that coincide with the Pareto optimal choice
more » ... re both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure. Quantum games as a field received a lot of attention from the early works of Meyer [1], and has grown steadily ever since. Connections exist between quantum games and various other fields, such as Bell non-locality [3] and quantum logic [2], to name a few. Various aspects of quantum games, including the role of entanglement and multiple player extensions, explored by different authors can be found in references [4] [5] [6] . A solution to the quantum prisoners' dilemma in which players have a Nash equilibrium that is Pareto optimal ignited interest in quantum games [7] . However, this initial formulation drew criticism because it dramatically restricted the strategy space of the players and did not persist under maximal entanglement if arbitrary quantum strategies were allowed [8] . In this work, we enlarge the Hilbert space in a minimal way in order to arrive at a Nash equilibrium (NE) that is Pareto-optimal with maximal entanglement. In addition to the two player qubits, we consider a resource qubit that the referee, or interrogator, has access to. In the three-qubit Hilbert space, one can introduce tripartite entanglement which will lead to a much richer Nash equilibrium structure. Interestingly, for tripartite entanglement close to maximum (approaching a W-state), we obtain Nash equilibria that coincide with Pareto optimal points. The bipartite entanglement between the players' qubits partially explains the structure. In addition, it is shown that there is no NE for a GHZ state, which has a fundamentally different type of entanglement [9] . We briefly review the standard formulation of the prisoners' dilemma.
doi:10.1007/s11128-018-1915-9 fatcat:6pvr3rvudzdnnn3ofgjworclea