We study the entanglement between the internal (coin) and the external (position) degrees of freedom in the dynamic and the static deterministic aperiodic quantum walks (QWs). For the dynamic (static) aperiodic QWs, the coin depends on the time (position) and takes two coins C ( α ) and C ( β ) arranged in the two classes of generalized Fibonacci (GF) and the Thue–Morse (TM) sequences. We found that for the dynamic QWs, the entanglement of three kinds of the aperiodic QWs are close to the maximal value, which are all much larger than that of the homogeneous QWs. Further, the first class of GF (1st GF) QWs can achieve the maximum entangled state, which is similar to that of the dynamic disordered QWs. And the entanglement of 1st GF QWs is greater than that of the TM QWs, being followed closely by the entanglement of the second class of GF (2nd GF) QWs. For the static QWs, the entanglement of three kinds of the aperiodic QWs are also close to the maximal value and 1st GF QWs can achieve the maximum entangled state. The entanglement of the TM QWs is between 1st GF QWs and 2nd GF QWs. However, the entanglement of the static disordered QWs is less than that of three kinds of the aperiodic QWs. This is different from those of the dynamic QWs. From these results, we can conclude that the dynamic and static 1st GF QWs can also be considered as maximal entanglement generators.
We study the entanglement between the internal (coin) and the external (position) degrees of freedom in the dynamic and the static deterministic aperiodic quantum walks (QWs). For the dynamic (static) aperiodic QWs, the coin depends on the time (position) and takes two coins C ( α ) and C ( β ) arranged in the two classes of generalized Fibonacci (GF) and the Thue–Morse (TM) sequences. We found that for the dynamic QWs, the entanglement of three kinds of the aperiodic QWs are close to the maximal value, which are all much larger than that of the homogeneous QWs. Further, the first class of GF (1st GF) QWs can achieve the maximum entangled state, which is similar to that of the dynamic disordered QWs. And the entanglement of 1st GF QWs is greater than that of the TM QWs, being followed closely by the entanglement of the second class of GF (2nd GF) QWs. For the static QWs, the entanglement of three kinds of the aperiodic QWs are also close to the maximal value and 1st GF QWs can achieve the maximum entangled state. The entanglement of the TM QWs is between 1st GF QWs and 2nd GF QWs. However, the entanglement of the static disordered QWs is less than that of three kinds of the aperiodic QWs. This is different from those of the dynamic QWs. From these results, we can conclude that the dynamic and static 1st GF QWs can also be considered as maximal entanglement generators.
We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRWs). Disorder is modeled by adding another random feature to QRWs, i.e., the quantum coin that drives the system's evolution is randomly chosen at each position and/or at each time step, giving rise to dynamic, fluctuating, or static disorder. The first one is position independent, with every lattice site having the same coin at a given time; the second has time- and position-dependent randomness; and the third one is time independent. We show for several levels of disorder that dynamic disorder is the most powerful entanglement generator, followed closely by fluctuating disorder. Static disorder is the less efficient entangler, being almost always less efficient than the ordered case. Also, dynamic and fluctuating disorder lead to maximally entangled states asymptotically in time for any initial condition, while static disorder has no asymptotic limit and, similarly to the ordered case, has a long-time behavior highly sensitive to the initial conditions.
Quantum walk (QW), the quantum mechanical counterpart of classical random walk, has recently been studied in various fields. The evolution of the discrete time quantum walk can be described as follows: the walker changes its spin state by the coin operator C, then takes one step left or right according to its spin state. For homogeneous quantum walk, the coin operator is independent of time and the standard deviation of the position grows linearly in time. It is quadratically faster than that in the classical random walk. In this work, we numerically study the dynamical behaviors of spreading in a one-dimensional discrete time quasiperiodic quantum walk (DTQQW). The DTQQW is that the coin operator is dependent on time and takes two different coins C() and C() arranged in generalized Fibonacci (GF) sequences. The GF sequences are constructed from A by the recursion relation: AAmBn, BA, for m, n are positive integers. They can be classified into two classes according to the wandering exponent . For 0, they belong to the first class, and for 0, they belong to the second class. For one dimensional system, the behaviors of two classes of GF systems are different either for the electronic spectrum of an electron in quasiperiodic potentials or for the quantum phase transitions of the quasiperiodic spin chains. In this paper, we discuss the cases of two different C operators (C();C()) arranged in GF sequences and find that the spreading behaviors are superdiffusion (the standard deviation of the position ~t; 0:5 1) for the two classes of GF DTQQW. For the second class of GF DTQQW, the exponent values are larger than those of the first class of GF DTQQW in the case of two identical C operators. By exploring the probability distribution in the real space, we find that for the first class of GF DTQQW, the probability distributions are almost the same for different initial states and are similar to the classical Gaussian distribution. For the probability distributions of the second class of GF DTQQW, there are two peaks at the two edges and the height of the two peaks can be different for different initial states. They are similar to the ballistic distribution of the homogeneous quantum walk. Therefore, we conclude that for the first class of GF DTQQW, the spreading behaviors are close to those of the classical random walk ( = 0:5) while for the second class of GF DTQQW, they are close to those of the homogeneous quantum walk ( = 1). This result is quite different from the characteristics of the quantum phase transitions in two classes of GF quasiperiodic quantum spin chains.
We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse, and Rudin-Shapiro. We use a generalized Hadamard coin, C[over ̂]_{H}, as well as a generalized Fourier coin, C[over ̂]_{K}. We verify the QW experiences a slowdown of the wave packet spreading, σ^{2}(t)∼t^{α}, by the aperiodic jumps whose exponent, α, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely, (1) while the superdiffusive regime (1<α<2) is predominant, α displays an unusual sensibility with the type of coin operator where the more pronounced differences emerge for the Rudin-Shapiro and random protocols and (2) even though the angle θ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of α with θ. Fingerprints of the aperiodicity in the hoppings are also found when distributional measures such as the Shannon and von Neumann entropies, the Inverse Participation Ratio, the Jensen-Shannon dissimilarity, and the kurtosis are computed, which allow assessing informational and delocalization features arising from these protocols and understanding the impact of linear and nonlinear correlations of the jump sequence in a quantum walk as well. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps.
Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.
We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the system's time evolution. We also show that maximal entanglement is achieved independently of the initial state of the walker, study the number of steps the system must move to be within a small fixed neighborhood of its asymptotic limit, and propose two experiments where these ideas can be tested.
We investigate the role of different aperiodic sequences in the dynamics of single quantum particles in discrete space and time. For this we consider three aperiodic sequences, namely, the Fibonacci, Thue-Morse, and Rudin-Shapiro sequences, as examples of tilings the diffraction spectra of which have pure point, singular continuous, and absolutely continuous support, respectively. Our interest is to understand how the order, intrinsically introduced by the deterministic rule used to generate the aperiodic sequences, is reflected in the dynamical properties of the quantum system. For this system we consider a single particle undergoing a discrete-time quantum walk (DTQW), where the aperiodic sequences are used to distribute the coin operations at different lattice positions (inhomogeneous DTQW) or by applying the same coin operation at all lattice sites at a given time but choosing different coin operation at each time step according to the chosen aperiodic sequence (time dependent DTQW). We study the energy spectra and the spreading of an initially localized wave packet for different cases, finding that in the case of Fibonacci and Thue-Morse tilings the system is superdiffusive, whereas in the Rudin-Shapiro case it is strongly subdiffusive. Trying to understand this behavior in terms of the energy spectra, we look at the survival amplitude as a function of time. By means of the echo we present strong evidence that, although the three orderings are very different as evidenced by their diffraction spectra, the energy spectra are all singular continuous except for the inhomogeneous DTQW with the Rudin-Shapiro sequence where it is discrete. This is in agreement with the observed strong localization both in real space and in the Hilbert space. Our paper is particularly interesting because quantum walks can be engineered in laboratories by means of ultracold gases or in optical waveguides, and therefore would be a perfect playground to study singular continuous energy spectra in a completely controlled quantum setup.