Thermodynamics of N-dimensional quantum walks

The entanglement between the position and the coin state of an N-dimensional quantum walker is shown to lead to a thermodynamic theory. The entropy, in this thermodynamics, is associated with the reduced density operator for the evolution of chirality, taking a partial trace over positions. From the asymptotic reduced density matrix it is possible to define thermodynamic quantities, such as the asymptotic entanglement entropy, temperature, and Helmholz free energy. We study in detail the case of a two-dimensional quantum walk, in the case of two initial conditions: a nonseparable coin-position initial state and a separable one. The resulting entanglement temperature is presented as a function of the parameters of the system and those of the initial conditions.