Discrete time quantum walk model for single and entangled particles to retain entanglement in coin space

摘要

In most widely discussed discrete time quantum walk model, after everyunitary shift operator, the particle evolves into the superposition of positionspace and settles down in one of its basis states, loosing entanglement in thecoin space in the new position. The Hadamard operation is applied to let theparticle to evolve into the superposition in the coin space and the walk isiterated. We present a model with a additional degree of freedom for theunitary shift operator $U^{\prime}$. The unitary operator with additionaldegree of freedom will evolve the quantum particle into superposition ofposition space retaining the entanglement in coin space. This eliminates theneed for quantum coin toss (Hadamard operation) after every unitarydisplacement operation as used in most widely studied version of the discretetime quantum walk model. This construction is easily extended to a multipleparticle quantum walk and in this article we extend it for a pair of particlesin pure state entangled in coin degree of freedom by simultaneously subjectingit to a pair of unitary displacement operators which were constructed forsingle particle. We point out that unlike for single particle quantum walk,upon measurement of its position after $N$ steps, the entangled particles arefound together with 1/2 probability and at different positions with 1/2probability. This can act as an advantage in applications of the quantum walk.A special case is also treated using a complex physical system such as, interspecies two-particle entangled Bose-Einstein condensate, as an example.