Abstract
The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.
Highlights
We find that analyzing quantum games using geometric algebra (GA) comes with some clear benefits, for instance, improved perception of the quantum mechanical situation involved and an improved geometrical visualization of quantum operations
We find that an improved geometrical visualization becomes helpful in significantly simplifying quantum calculations, for example unitary transformations on a single qubit become rotations of a vector as displayed on the Bloch sphere, and two qubits can be modeled in a real SO(6) space [67] and we find unique expressions in GA, such as Eq (9) describing measurement outcomes for N qubits
We find that by using an EPR type setting we produce a faithful embedding of symmetric mixed-strategy versions of classical twoplayer two-strategy games into its quantum version, and that GA provides a simplified formalism over the field of reals for describing quantum states and measurements
Summary
Its origins can be traced to earlier works [1,2,3,4], the extension of game theory [5,6] to the quantum regime [7] was proposed by Meyer [8] and Eisert et al [9] and has since been investigated by others [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. At these quantum NE the players can have higher payoffs relative to what they obtain at the NE in the mixed-strategy version of the classical game.
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