Role of two-qubit entangling operators in the modified Eisert-Wilkens-Lewenstein approach of quantization

One of the central ideas in the theory of quantum games is to quantize the classical games in a proper mathematical and physical framework. In order to understand the prowess of quantumness, it is ensured that classical game is the subset of quantum game. This has been stressed and established in Eisert, Wilkens and Lewenstein (EWL) quantization scheme, by imposing a commutation condition between the entangling operators and the pure strategies of the game. In the present work, we modify the EWL scheme by relaxing this condition, with an aim to explore the significance of two-qubit entangling operators. We establish two results: firstly, conversion of symmetric to potential game under suitable conditions of initial states and strategies. Secondly, non-conversion of a zero-sum game to nonzero-sum game irrespective of initial states and strategies. Since these results are shown to be consistent with that of Marinatto and Weber quantization scheme, we justify the equivalence of two quantization schemes.