  We study, from a thermodynamic perspective, the equilibrium states of a qubit
interacting with an arbitrary environment of dimension N>>2. We show that even
in presence of memory about the initial state, in some cases the qubit can be
considered in a thermal state characterized by an entanglement Hamiltonian,
which encodes the effects of the environment, and an initial-state- dependent
entanglement temperature that measures the degree of entanglement generated
between the system and its environment. Geometrical aspects of the thermal
states are studied, and the results are confirmed for the concrete case of the
Quantum Walk on the Line.
