Stationary Measures Of Space-Inhomogeneous Three-State Quantum Walks

As quantum analogs of classical random walks(RWs), quantum walks(QWs) have been used widely in quantum computing, quantum information, quantum prob-ability and biophysical systems. In the past fifteen years, QWs have attracted much interest of many researchers both in mathematics and in physics. One hot research topic is the asymptotic behavior of discrete-time quantum walks(DTQWs), which includes the localization behavior and the ballistic spreading behavior. The station-ary measure of DTQWs plays an important role in understanding the asymptotic behavior of DTQWs. Many results have been obtained of the space-inhomogeneous two-state QWs. The degenerated eigenvalues of three-state QWs have quite differ-ent degrees of freedom from those of two-state QWs. In this paper, we consider the stationary measure of the space-inhomogeneous three-state QWs. Our main work is as follows:We investigate a space-inhomogeneous three-state quantum walk on the line, which we call the three-state Wojcik walk. We calculate its eigenvalues, and by using the SGF method introduced by Konno et al., we obtain its stationary measure. We find that the measure decays exponentially with respect to position under some mild conditions; however, if the walk takes -1 as its eigenvalue, the asymptotic behavior of the measure is independent of position, which contrasts sharply with that of two-state quantum walks. On the other hand, it is known that the Grover walk is a space-homogeneous three-state quantum walk. In a recent paper, Konno actually showed that the stationary measure of the Grover walk decays exponentially with respect to position and, in particular, the exponential decay rate is just 49-20(?)= 0.010205…. We find that for the three-state Wojcik walk, its stationary measure can decay exponentially with respect to position and, in particular, its exponential decay rate can be 49-20(?).We also consider a two-phase three-state QW with one defect on the line. Sim-ilarly, by using the SGF method, we calculate its eigenvalues and get its stationary measure. Quite like the case of three-state Wojcik walk, the stationary measure of the two-phase three-state QW decays exponentially with respect to position under some mild condition; if the walk takes -1 as its eigenvalues, the asymptotic behav-ior of the measure is independent of position which contrasts sharply with that of two-phase two-state QWs.