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Virtual temperatures as a key quantifier for passive states in quantum thermodynamic processes

We analyze the role of virtual temperatures for passive quantum states through the lens of majorization theory. A mean temperature over the virtual temperatures of adjacent energy levels is defined to compare the passive states of the system resulting from isoenergetic and isoentropic transformations. The role of the minimum and the maximum (min-max) values of the virtual temperatures in determining the direction of heat flow between the system and the environment is argued based on majorization relations. We characterize the intermediate passive states in a quantum Otto engine using these virtual temperatures and derive an upper bound for the Otto efficiency that can be expressed in terms of the min-max virtual temperatures of the working medium. An explicit example of the coupled-spins system is worked out. Moreover, virtual temperatures serve to draw interesting parallels between the quantum thermodynamic processes and their classical counterparts. Thus, virtual temperature emerges as a key operational quantity linking passivity and majorization to the optimal performance of quantum thermal machines.