Entanglement is an indispensable resource for performing various quantum tasks (see [1] for a recent review, [2] and reference therein for entanglement preparation). Several schemes have been proposed [3-6] for the generation and distribution of entanglement between different systems, most of which involve an initial entangling of the two systems followed by spatial separation. Such spatially separated and entangled states can be used for quantum communication protocols, for example, quantum cryptography [7] and quantum teleportation [8]. Amount of entanglement degrades with increase in spatial separation because of physical limitations and noise effect. One way of circumventing this problem would be to generate entanglement when the two systems are spatially separated.

In this article, we present a new scheme to efficiently generate entanglement between two spatially separated systems from a single particle system [9]. It has been shown by two of the present authors that a quantum walks evolution of a particle in a one-dimensional lattice results in the entanglement of the lattice sites after a sufficient number of walk steps [10]. Although this spatial entanglement by itself does not have much physical significance, we nevertheless make use of it in entangling two initially unentangled systems that are spatially separated. Direct control over the quantum coin operation makes it possible to control the dynamics of the evolution of the quantum walk [11,12] which in turn allows us to optimize the entanglement output. The ability to localize the evolution at different lattice sites simultaneously, a novel phenomenon which has been discussed for the first time in this article, leads to a substantial improvement in the entanglement generated. This is a generic scheme that can be implemented to entangle any two systems that interact with the modes entangled due to quantum walk. Experimental implementation of quantum walk has been reported with samples in nuclear magnetic resonance (NMR) systems [13-15]; in the form of optical Galton board [16] and quantum quincunx [17]; in the continuous tunneling of light fields through waveguide lattices [18]; in the phase space of trapped ions [19, 20]; with single optically trapped atoms [21]; and with single photon [22,23]. There are various other schemes proposed to implement quantum walk in other systems [24-26]. Using our scheme, all these systems have the potential to generate entanglement between two spatially separated, uncorrelated systems.

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 Figure 1). First, a photon in the initial state is passed through the beam splitter. The state of the photon passing through the beam splitter are spatially separated into the horizontal(h) and vertical(v) modes and it can be written as

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 Figure 1. As we commented at the end of the Introduction, the new scheme we propose below has the ingredients of the preceding toy model; which we now describe.

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 [12] that complete control over the dynamics of the walk can state at the new position into a superposition of the be accomplished by choosing C as a three parameter SU(2) group element; here, however, we take, for simplicity, the widely used Hadamard operator, that is,

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. [29]). The next task is to use this spatial entanglement to generate entanglement between two uncorrelated systems which we will call A and B in the following.

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 [30]. The protocol, we propose, to generate entanglement between systems A and B is to: (1) evolve the desired number of quantum walk steps on a particle in lattice to generate entanglement between two spatial modes without having A and B interacting with the lattice, and (2) stop the quantum walk evolution and switch on the interaction between A and B at the desired lattice sites which are spatially entangled. The interaction of systems A and B with the entangled lattice sites depends upon the nature of the system, on which quantum walk is being performed and the properties of the system A and B. In this section, we will consider entangling two distant spins, fermions using quantum walk on a one-dimensional lattice consisting of spin-1/2 particles and this can be extended to bosonic system as shown in Section 6.

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 [31] a spin-1/2 system can be mapped to a spinless fermionic system. Therefore, a one-dimensional spin-1/2 lattice with all spins but one, pointing downward can be viewed as a system consisting of a single spinless fermion. By attaching an extra coin degree of freedom with the fermion one can perform quantum walk. Now if systems A and B are also spin-1/2 particles, spanned by the basis interacting with the spins at ±l lattice sites, the interaction Hamiltonian can be written as

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. [32] where quantum walk has been used for state transfer in a spin system.

In Figure 2, we show the evolution of entanglement between systems and for different initial states, using concurrence [33] as the measure. Following the previous section, we have considered the interaction of and with the −1 and +1 lattice sites, respectively, after three steps of the walk. We observe that the amount of entanglement depends on the initial state of the system and different

, (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 [10]: 1) the degree of interference during the quantum walk, and 2) the value of the probability amplitude at the particular lattice sites. Both of these can be achieved by evolving and localizing the quantum walk around the desired lattice sites with some degree of interference. To further understand the contribution of localization of quantum walk to increase the degree of spatial entanglement, we will consider the lattice state after t steps of quantum walk that can be written as

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 [12]. Even for θ close to 0 and π/2 the interference effect will be very small resulting in very small or zero spatial entanglement. When θ = π/24 the walk evolves with good degree of interference but with a spread of the amplitudes in position space (asymptotically, the distribution is homogeneous) resulting in a very low amplitude for lattice sites ±l far away from each other. Therefore, to maximize the amount of spatial entanglement between lattice sites irrespective of the separation distance, one needs to control the quantum walk evolution in such a way that its amplitude is localized, with a good degree of interference for any lattice site separation distance.

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 [12]. Therefore, even for very large t steps, of walk delocalize at ±l position with very small. In Ref. [36] it was shown that choosing a value of θ randomly picked from the interval {π/4, π/2} at each step of the walk, localizes the quantum walk distribution around the initial position and the localization is a result of interference. Therefore, choosing θ randomly from the interval {π/4, π/2} for each step of the after the walk is delocalized at ±l results in some improvement of at the cost of which in turn contributes for spatial entanglement.

Figure 3, shows the localization of the amplitudes at positions −95 and +95 after 100 and 200 steps of the walk; for the first 95 steps we choose θ ≈ π/36 followed by θ Î{π/4, π/2} for remaining steps. Note that the peaks do not move away even after 200 steps of the walk. This protocol for localization at desired sites will hold even for a very large spatial separation of lattice sites making it scalable. One should avoid using θ = 0 during the first part of the evolution, because as commented above, the walk, in this case, evolves without interference. As for the entanglement transferred to the uncorrelated systems A and B (see the previous section), in Figure 4, we show the evolution of their concurrence when they interact with lattice sites ±95 after 200 steps of the walk. Although the amount of entanglement in this case is smaller than that of Figure 2, qualitatively both evolutions are identical. The decrease in amount of entanglement is mainly due to the delocalization and localization of the walk which was not required for a 3-step walk used to obtain Figure 2.