A dynamical system induced by quantum walk

Abstract

We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an ℓ ∞ -infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportion to the number of edges of this graph. Furthermore when we add more tails (e.g., r-tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is (r + 1). *

Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

“…The Grover walk with tail was introduced by [16,17] in terms of the scattering theory. If we set some appropriate bounded initial state so that the support is included in the tail, the existence of the fixed point of the dynamical system induced by the Grover walk with tails is shown, and the stable generalized eigenspace H s , in which the dynamical system lives, is orthogonal to the centered generalized eigenspace H c [18] at every time step [19]. The centered generalized eigenspace is generated by the generalized eigenvectors of the principal submatrix of the time evolution operator of the Grover walk with respect to the internal graph, and all the corresponding absolute values of the eigenvalues are 1.…”

Dressed Photon Constant as a Key Parameter for the Conformal Cyclic Cosmology of Twin Universes

“…Equation (26) tells us that the Riemann curvature associated with these light fields takes the form R λρµν = F λρ F µν (= S λρ S µν ). In addition to R ν ν = 0, we can readily show R µν R µν = 0 using (23). Under the former condition R ν ν = 0, the Weyl tensor W λρµν assumes the form…”

“…We believe that the DP is generated through this pair annihilation of the Majorana field. Since the DP field is basically electromagnetic, once it is generated, its behavior in a uniform environment can be walk models [22][23][24]. Intuitively, however, we can expect that some kind of ζ-function enters here as the carrier to convey the above singularity waves, which explains the observation of ζ-function singularities in the quantum walks.…”

“…where (•) * denotes the complex conjugate of (•). The right-hand side of (39) can be evaluated by the lightlike case of the CD field (23), in which we haveT µν = ρC µ C ν . For the lightlike case, we have…”

Quantum Walk and Dressed Photon

“…Although the term "jellyfish graph" itself is introduced in this paper, Feldman and Hillery already considered the discrete time quantum walk on this kind of graph [4] and continuous time quantum walk had been considered by Farhi and Gutmann [3]. Higuchi and Segawa [6] also study this type of graph and showed the following theorem (We modify the statement with our terminology). Theorem 8 Consider a quantum walk on jellyfish graph being free on each half-line l (i) , i.e., a quantum walk satisfying the condition that for any arrow a in l (i) and any arrow a ′ in l (i)…”