Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators

Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outcome of an accidental measurement of its chirality.

Figure 1

The two components of the wave function at $t=31$. The red dots correspond to ${\psi }_{+}^{}(n,t)$, whereas the blue diamonds mark the values of ${\psi }_{-}^{}(n,t)$.

Figure 2

The two components of the wave function at $t=100$, for $p=0.75$. The red dots denote ${\psi }_{+}^{}(n,t)$, whereas the blue diamonds designate the values of ${\psi }_{-}^{}(n,t)$.

Figure 3

Evolution of the entropy of entanglement as a function of time for $p=0.5$, green boxes, and $p=0.75$, magenta triangles. Observe how the entropy decreases monotonically in both cases.

Figure 4

Decay of the entropy of entanglement as a function of time. The green solid line corresponds to $p=0.5$, whereas the magenta dashed line shows the behavior of the entanglement when $p=0.75$. The black dotted line ($\sim \left[{log}_{2}4t\right]/4t$) serves as a guide for the eye.

Figure 5

Probability mass function $\rho (n,t)$ at $t=100$, for $p=0.75$. Red dots were obtained from the wave function of the coherent process, whereas the black dashed line corresponds to a Gaussian approximation, Eq. (61).

Figure 6

Probability mass function $\rho (n,t)$ for $t=100, p=0.5$, and different values of the measurement probability $q$. Data were obtained by averaging 1 000 000 different realizations of the process, except for the case $q=0.00$ that corresponds to the coherent walk. Labels are ordered to indicate increasing variance.