Asymptotic reduced density matrix of discrete-time quantum walks

Abstract

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 P0 projector, reduced density matrix, can be obtained by T r1 (P0 ⊗ I C) or equivalently T r2 (I ⊗ P0 C). It is worth to mention that characteristic matrix C is independent of the initial state and just depends on coin operator, so by finding this matrix for specific type of QW the long-time behavior of it, such as local state of the coin after a long time walking and asymptotic entanglement between coin and position will be completely known for any initial state. We have found the characteristic matrix C for general coin operator, U (2), as well as exact form of this matrix for local initial state.