A 2000 paper by Nayak and Vishwanath provides an analysis of the dynamics of quantum random walks. In this paper, they mention a "naive" approach to defining a walk. I include the quote as follows:
In direct analogy, one may naively try to define quantum walk on the line as follows: at every time step, the particle moves, in superposition, both left and right with equal amplitudes (perhaps with a relative phase difference). However, such a walk is physically impossible, since the global process is non-unitary.
Intuitively, I imagine this is the case because, without further qualification, it seems that the various probability amplitudes would interfere constructively and lead to a wavefunction that was not normalized. Although it still seems like it might be possible to induce a relative phase difference such that normalization is achieved.
How does one prove/verify the author's statement in a formal way that would be satisfying to theorists?