Localization of two-particle quantum walk on glued-tree and its application in generating Bell states

Abstract

Studies on two-particle quantum walks show that the spatial interaction between walkers will dynamically generate complex entanglement. However, those entanglement states are usually on a large state space and their evolutions are complex. It makes the entanglement states generated by quantum walk difficult to be applied directly in many applications of quantum information, such as quantum teleportation and quantum cryptography. In this paper, we firstly analyse a localization phenomena of two-particle quantum walk and then introduce how to use it to generate a Bell state. We will show that one special superposition component of the walkers' state is localized on the root vertex if a certain interaction exists between walkers. This localization is interesting because it is contrary to our knowledge that quantum walk spreads faster than its classical counterpart. More interestingly, the localized component is a Bell state in the coin space of two walkers. By this method, we can obtain a Bell state easily from the quantum walk with spatial interaction by a local measurement, which is required in many applications. Through simulations, we verify that this method is able to generate the Bell state $$\frac{1}{\sqrt{2}}(|A \rangle _1|A\rangle _2 \pm |B\rangle _1|B\rangle _2)$$12(|Aź1|Aź2±|Bź1|Bź2) in the coin space of two walkers with fidelity greater than $$99.99999\,\%$$99.99999% in theory, and we have at least a $$50\,\%$$50% probability to obtain the expected Bell state after a proper local measurement.