Chirality asymptotic behavior and non-Markovianity in quantum walks on a line

We investigate the time evolution of the chirality reduced density matrix for a discrete-time quantum walk on a one-dimensional lattice. The matrix is obtained by tracing out the spatial degree of freedom. We analyze the standard case, without decoherence, and the situation in which decoherence appears in the form of broken links in the lattice. By examining the trace distance for possible pairs of initial states as a function of time, we conclude that the evolution of the reduced density matrix is non-Markovian, in the sense defined by Breuer, Laine, and Piilo [Phys. Rev. Lett. 103, 210401 (2009)]. As the level of noise increases, the dynamics approaches a Markovian process. The highest non-Markovianity corresponds to the case without decoherence. The reduced density matrix tends always to a well-defined limit that we calculate, but only in the decoherence-free case is this limit nontrivial.

Figure 1

Left panel: Asymptotic trace distance as a function of the angles $\gamma$ and $\phi$, representing the initial conditions of ${\rho }_2$. The initial conditions of ${\rho }_1$ are given by $\gamma =0$. Right panel: The contour levels corresponding to the left panel are mapped to the Bloch sphere, using the same color convention.

Figure 2

Same as Fig. 1, but now the initial conditions for ${\rho }_1$ are given by $\gamma =\pi /4$ and $\phi =\pi$.

Figure 3

Trace distance, as a function of the number of time steps, between the whole density matrices (dashed lines) and between the corresponding reduced density matrices (solid lines). The initial state ${\rho }_1\left(0\right)$ is defined by Eq. (9) with $\gamma =0$, while ${\rho }_2\left(0\right)$ with $\gamma =\pi$. Different values of the decoherence parameter $p$ have been considered.

Figure 4

Contribution to the non-Markovianity measure, as a function of the number of time steps, evaluated for the pair of initial states ${\rho }_1\left(0\right)$ and ${\rho }_2\left(0\right)$ that maximizes the integral in Eq. (53). Different values of the decoherence parameter $p$ have been considered.

Figure 5

Contribution to the non-Markovianity measure, as a function of the decoherence parameter, calculated for the pair of initial states ${\rho }_1\left(0\right)$ and ${\rho }_2\left(0\right)$ that maximizes the integral in Eq. (53), and for the time interval $[0,200]$.