Initial-state-dependent thermalization in open qubits

We study, from a thermodynamic perspective, the equilibrium states of a qubit interacting with an arbitrary environment of dimension $N\gg 2$. We show that even in the 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.

Figure 1

Level plots of the entanglement temperature over the Bloch sphere for a time-averaged RDO whose Kraus operators are orthogonal projectors. The temperature $T_0$ (green point) corresponds to the asymptotic limit of all initial pure states located on the intersection of the plane $\mathrm{Re}\left(\kappa \right)x-\mathrm{Im}\left(\kappa \right)y+z=\sqrt{1+|\vec{\kappa }{|}^2}cos{\alpha }_0$ with the sphere, as shown by the arrows. Also represented are the minimum temperature states $T=0$ (blue points) and the $T=\infty$ state (red point).

Figure 2

Level plots of the entanglement temperature on the Bloch sphere, for an initially distributed Gaussian walker, in the limit $\xi \gg 1$. The initial coin states on the equator (red circumference, $\alpha =\pi /2$) thermalize at $T=\infty$ (red point at the center of the Bloch sphere), those in the intermediate latitude (green circumference, $0<\alpha <\pi /2$) converge to the green state at its center at some finite temperature, while the blue states at the poles, corresponding to $\alpha =1$, remain at $T=0$ during the evolution.

Figure 3

Level plots of the entanglement temperature in the ($\gamma ,\phi$) space, for a Hadamard walk with an initially Gaussian distributed walker. The thick central line corresponds to $T=\infty$, while the center of the closed lines is associated to the states that thermalize at $T=0(\gamma =\pi /4,\phi =0)$ and $(\gamma =3\pi /4,\phi =\pi )$.