Institute of Fundamental Technological Research

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

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.

Quantum walks, Entanglement, Gaussian states

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.

Quantum walks,Quantum entanglement,Entanglement production