Quantum walk on a line with two entangled particles

We introduce the concept of a quantum walk with two particles and study it for the case of a discrete time walk on a line. A quantum walk with more than one particle may contain entanglement, thus offering a resource unavailable in the classical scenario and which can present interesting modifications on quantum walks with single particles. In this work, we show both numerically and analytically how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, the probability to find at least one particle in a certain position after $N$ steps of the walk, as well as the average distance and the squared distance between the two particles, can be larger or smaller than the case of two unentangled particles, depending on the initial conditions we choose. This resource can then be tuned according to our needs to modify the features of a quantum walk. Experimental implementations are briefly discussed.

Figure 1

(Color online) Probability distribution for a classical random walk (a) on a line after $N=100$ steps, as well as for a quantum walk (b) with initial state $\mid 0⟩\otimes \mid \uparrow ⟩$ and a Hadamard coin.

Figure 2

(Color online) Two-particle probability distributions after $N=30$ steps for different initial conditions: (a) separable state ${\mid {\psi }_0^S⟩}_{12}$, (b) ${\mid {\psi }_0^{+}⟩}_{12}$ state, and (c) ${\mid {\psi }_0^{-}⟩}_{12}$ state. Note the different vertical ranges.