Asymptotic reduced density matrix of discrete-time quantum walks

In this article we show that for any quantum walker with m-dimensional coin subspace, we have m ² × m ² specific constant matrix $C$ where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P ₀ projector, the reduced density matrix can be obtained by $T{r}_1(P_0\otimes IC)$ or equivalently $T{r}_2(I\otimes P_{0}C)$. It is worth mentioning that the characteristic matrix $C$ is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix $C$ for the general coin operator, $U\left(2\right)$, as well as the exact form of this matrix for the local initial state. (paper)