The envelope of one-dimensional discrete-time quantum walk

A mathematical model is presented for a one-dimensional discrete-time quantum walk, which is initiated from a quantum initial state and governed by a coin operator. When the coin operator is a flip operator, a path analysis formula is employed to compute the position probability distribution. For a general coin operator, matrix decomposition is utilized to transform it into the equivalent flip operator. When a walker undergoes $n$ steps of evolution, it is observed that the probability of the walker occupying any given position exhibits the existence of both maximum and minimum values, irrespective of the quantum initial state. By linking these extreme positions together, a confined region is delineated, the boundary of which is designated as the envelope of the quantum walk. Remarkably, the envelope is independent of the quantum initial state. To facilitate the computation of this envelope, the relevant formulas are transformed into rational expressions, wherein both the numerators and denominators are represented by polynomials with even integer coefficients. These polynomials are classified to determine the coefficients of the numerator polynomials. An analysis is conducted to identify the location of the maximum value of the envelope, thereby examining the maximum value of the position probability distribution.