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We present a scheme for generating entanglement between two spatially separated systems from the spatial entanglement generated by the interference effect during the evolution of a single-particle quantum walk. Any two systems which can interact with the spatial modes entangled during the walk evolution can be entangled using this scheme. A notable feature is the ability to control the quantum walk dynamics and its localization at desired pair lattice sites irrespective of separation distance resulting in a substantial control and improvement in the entanglement output. Implementation schemes to entangle spatially separated atoms using quantum walk on a single atom is also presented.

Entanglement is an indispensable resource for performing various quantum tasks (see [

In this article, we present a new scheme to efficiently generate entanglement between two spatially separated systems from a single particle system [

This article is arranged as follows. In Section 2 we describe a toy model which has the basic ingredients of our proposal: 1) two entangled modes are generated and distributed to the distant locations of two uncorrelated systems A and B; 2) the entanglement of these modes is then transferred to A and B via some interaction. In Section 3 we describe the discrete-time quantum walk model and the entanglement between its spatial degrees of freedom. Section 4 discusses how to use this spatial entanglement to entangle two uncorrelated, spatially separated systems A and B. The Hamiltonian modelling the interaction of these systems with the lattice sites is motivated by two examples: quantum walk with single photons and quantum walk in a spin chain; in both cases A and B are taken to be two-level systems. In Section 5 we explain how to localize the quantum walk distribution around desired lattice sites, in such a way that the entanglement between these sites is maximized. We then show how this affects the entanglement transferred to systems A and B. In Section 6 we propose experimental implementtations of our proposal to entangle two uncorrelated atoms in an optical lattice. We conclude in Section 7.

Before proceeding to our scheme, we will introduce the basic idea using a simple model involving a beam splitter, a photon, and two two-level atoms. The aim is to generate entanglement between the uncorrelated atoms, labeled by A and B, which are placed in distant locations (see

such that, where and represent the probability of finding the photon in the h and v modes, respectively. For convenience, we can rewrite the state of the photonic modes in terms of the number of photons in each polarization mode:

The state represents one photon in the h(v) mode and no photon in v(h) mode. This state is entangled unless α or β is zero. This entanglement between the polarization modes can be used to entangle A and B. This is done by placing atoms A and B initially in the ground state at the two exit points of the photon coming from the beam splitter. The conditions are such that if the photon is in v(h) mode, atom A(B) will get excited, that is,. The final collective state of these two atoms can be written as:

This provides a very simple model of generating entanglement between two distant systems from the entanglement between the photonic modes. Its pictorial representation is as in

Discrete-time quantum walk is defined on a coin Hilbert space Η_{c} and position Hilbert space Η_{P}. In one dimension, Η_{c} is spanned by the basis states and and Η_{P}. is spanned by the basis state. Each step of the quantum walk on a particle initially in superposition of the coin states at origin (j = 0) given by,

is implemented by applying a conditional shift operation S followed by the quantum coin operation C. The operation S can be defined such that the state moves to the left (right),

The operation C follows the operation S and evolves the coin basis states [11,27,28]. It was shown in [

which is an element of U(2) group. In order to show the analytics of spatial entanglement, we consider only three steps of the walk, after which the state of the particle can be written as:

where

and. Let us concentrate on the lattice sites −1 and +1 only, and denote its position states as and. The reduced density matrix, after tracing out the other lattice sites and the coin degrees of freedom, is:

where and refers to the state when the walker is neither in the +1 nor in the −1 site. The partial transpose of the above matrix will always be nonpositive. Therefore, the reduced density matrix represents an entangled state showing that the lattice sites −1 and +1 are entangled (cf. [

It was shown some years ago that two distant spins A and B become entangled after interacting with the spins of an entangled pair through a beam-splitter-like Hamiltonians [

For a spin-1/2 system, the spin hops from one lattice site to another in a quantum walk evolution. By using Jordon-Wigner transformation [

In the preceding Hamiltonian, and stands for the lowering, and raising, operator for spin respectively. After letting A interact with spatial mode of spin at –l and B with spatial mode of spin at –l for time t given by the evolution operator

and the state of the system AB can be written as:

Here are Kraus operators and:

is an orthonormal basis in

forms the set of eigenvalues and eigenvectors for and ρ_{AB} is the initial state of the system AB. The unitary operator W, responsible for the joint evolution is:

where P is a permutation operator such that

More detailed description of the process of transferring the entanglement in spin system can be seen in Ref. [

In

, (b) and (c).

separable states achieve maxi-mum entanglement at the same time.

The degree of spatial entanglement between two lattice sites depends largely on two points [

represents a state where lattice site j is occupied with other sites being empty and represents the corresponding coin state. Let us say we are interested in particular lattice sites ±l. Then in the reduced density matrix which is of the form given by Equation (8), with off diagonal terms and diagonal terms depending on the amplitude of all states, the states other than will also contribute to If the amplitude of all the states except for is zero then the coefficient of will be zero resulting in the maximum value of which contributes for spatial entanglement. Therefore, in localized states, that is, the states in which the amplitude is localized in a very narrow lattice space, the coefficient of will be very small resulting in maximizing and hence the amount of spatial entanglement will be more.

To realize this using quantum walk, we will begin by generalizing the previous discussions on quantum walk evolution by taking a more general coin operator

During the walk evolution, if θ = 0 the amplitude of the two basis states move away from each other and for θ = π/2 the amplitude shifts between the origin (j = 0) and its neighboring positions (±1). In both these cases the walk evolves without resulting in any interference [

Localization of quantum walk at the origin has been discussed in Refs. [34-38]. Here, we briefly discuss a way to localize the walk around the desired lattice sites such that, it is scalable for site ±l far away from each other. This can be done by first delocalizing the walk to sites ±l with minimum interference resulting in large and very small followed by localization around sites ±l to improve at the cost of. As discussed earlier, choosing θ very close to zero will result in two peaks at lattice sites ±l = ±tcos(θ) moving away from each other with minimal interference, where t is the number of steps of the walk [