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arXiv:1807.01361 · Physical Review A

Connecting velocity and entanglement in quantum walks

Alexandre C. Orthey, Edgard P. M. Amorim

Published Mar 14, 2019• 18 citations
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Abstract

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 $\ensuremath{\delta}$-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.

DOI
10.1103/physreva.99.032320

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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 \ensuremathδ\ensuremath{\delta}\ensuremathδ-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 investigate the relation between the ballistic spreading 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 long-time expected position together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while the expected position with the quantum coin phases give it for highly delocalized states.
We study the entanglement between the internal (spin) and external (position) degrees of freedom of the discrete time quantum walk on the line. We show the dependence between the asymptotic entanglement and the initial dispersion for delocalized states and establish a way to connect the asymptotic entanglement between local and delocalized states. We find out a simple relation between the phases of the initial spin state, which evidences that the delocatization of the state enhances from one (local state) to a continuous set of initial spin states reaching maximal entanglement. Our main finding could be tested in current experimental platforms.
We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-1/21/21/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
We report a systematic study of asymptotic behavior of entanglement between the position and coin degrees of freedom for a one-dimensional discrete quantum walk. Effects of coin bias, initial coin asymmetry and phase difference and non-locality of initial position state on the fluctuations and asymptotic value of entanglement are investigated by using Fourier approach. Fluctuations in entropy are found to die out as power law in time as t-α with α = 1/2,3/2 and 5/2 depending on the initial state parameters of the system.
We investigate the spreading behavior of the one-dimensional discrete time quantum walk starting from local and Gaussian states. In order to compare both cases, we obtain analytically the average variance of the position by averaging over all spin states. We find out that the average variance for the local case is independent on the quantum coin. However, when the quantum walk starts from a Gaussian state, its average variance depends on the sum of relative phases between spin states given by the quantum coin, and also, on the initial dispersion of the state. We perform numerical simulations of the average probability and variance along the time to confront them with our analytical results.