We study the entanglement between the internal (spin) and external (position) degrees of freedom of the one-dimensional discrete time quantum walk starting from local and delocalized initial states whose time evolution is driven by Hadamard and Fourier coins. We obtain the dependence of the asymptotic entanglement with the initial dispersion of the state and establish a way to connect the asymptotic entanglement between local and delocalized states. We find out that the delocalization of the state increases the number of initial spin states which achieves maximal entanglement from two states (local) to a continuous set of spin states (delocalized) given by a simple relation between the angles of the initial spin state. We also carry out numerical simulations of the average entanglement along the time to confront with our analytical results.Comment: One column, 15 pages, 6 figure

That quantum correlations can be generated over time between the spin and the position of a quantum walker is indisputable. The creation of bipartite entanglement has also been reported for two-walker systems. In this scenario, however, since the global state lies in a fourpartite Hilbert space, the question arises as to whether genuine multipartite entanglement may develop in time. Also, since the spatial degrees of freedom can be viewed as a noisy channel for the two-spin part, one may wonder how other nonclassical aspects (quantumnesses), such as Bell nonlocality, Einstein-Podolsky-Rosen steering, quantum discord, and symmetrical quantum discord, evolve in time during the walk. The scarcity of such a broader investigation is possibly due to computational difficulties associated with the recursive nature of quantum walks. Here, we work around this issue by introducing a simplified Gaussian model which proves to be very accurate within a given domain and powerful for the analytical studies. Then, for an instance involving two noninteracting quantum walkers, whose spins start in the singlet state, we quantify the aforementioned quantumnesses as a function of time, and evaluate violations of both realism and related aspects of locality. In addition, we analyze situations in which the initial two-spin state is affected by white noise. The typical scenario found is such that while genuine fourpatite entanglement increases over time, all the investigated quantumnesses vanish (suddenly or asymptotically) except realism-based nonlocality. Moreover, realism is prevented for all finite times.

We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following a delta-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while this velocity with the quantum coin phases give it for highly delocalized states.

We find out a few ways to improve the realization of entanglement between the internal (spin) and external (position) degrees of freedom of a quantum particle, through the insertion of disordered time steps in a one-dimensional discrete time quantum walk in different scenarios. The disorder is introduced by a randomly chosen quantum coin obtained from a uniform distribution among infinite quantum coins or only between Hadamard and Fourier coins for all the time steps (strong disorder). We can also decrease the amount of disorder by alternating disordered and ordered time steps throughout a quantum walk or by establishing a probability p < 0.5 to pick a Fourier coin instead of a Hadamard one for each time step (weak disorder). Our results show that both scenarios lead to maximal entanglement outperforming the ordered quantum walks. However, these last scenarios are more efficient to create entanglement, because they achieve high entanglement rates in fewer time steps than the former ones. In order to compare distinct disordered cases, we perform an average entanglement by averaging over a large set of initial qubits over time starting from one site (local state) or spread over many neighbor positions following a Gaussian distribution. Some transient behaviors from order to disorder in quantum walks are also evaluated and discussed. Experimental remarks based on available experimental platforms from the literature are made.