Asymptotic entanglement in 2D quantum walks

Abstract

The evolution operator of a discrete-time quantum walk involves a conditional\nshift in position space which entangles the coin and position degrees of\nfreedom of the walker. After several steps, the coin-position entanglement\n(CPE) converges to a well defined value which depends on the initial state. In\nthis work we provide an analytical method which allows for the exact\ncalculation of the asymptotic reduced density operator and the corresponding\nCPE for a discrete-time quantum walk on a two-dimensional lattice. We use the\nvon Neumann entropy of the reduced density operator as an entanglement measure.\nThe method is applied to the case of a Hadamard walk for which the dependence\nof the resulting CPE on initial conditions is obtained. Initial states leading\nto maximum or minimum CPE are identified and the relation between the coin or\nposition entanglement present in the initial state of the walker and the final\nlevel of CPE is discussed. The CPE obtained from separable initial states\nsatisfies an additivity property in terms of CPE of the corresponding\none-dimensional cases. Non-local initial conditions are also considered and we\nfind that the extreme case of an initial uniform position distribution leads to\nthe largest CPE variation.\n