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.

Figure 1

Eigenvalues of the reduced density matrix [Eqs. (80), (61), and (82)] as a function of the parameter $x=sin\gamma cos\phi$, with $\theta =\pi$. ${\Lambda }_1$, solid line; ${\Lambda }_2$, dashed line; and ${\Lambda }_3$, dot-dashed line.

Figure 2

Entanglement temperature [see Eq. (84)] as function of the dimensionless parameter $x=sin\gamma cos\phi$, with $\theta =\pi$.

Figure 3

Isotherms on the Bloch sphere. $|Z_{+}\rangle$ and $|Z_{-}\rangle$ are the North and South Pole, respectively. The two black points (“cold points,” corresponding to $T=0$) on the sphere are the points $\frac{1}{\sqrt{2}}\left(\right|Z_{+}\rangle +|Z_{-}\rangle )$ and $\frac{1}{\sqrt{2}}\left(\right|Z_{+}\rangle -|Z_{-}\rangle )$. The light (yellow) zone is the “hot zone,” $T=T_0$.

Figure 4

Entanglement temperature [see Eq. (93)] as a function of the dimensionless parameter $\gamma$.