Multidimensional quantum walks: Diabolical points, optical wave-like propagation, and multipartite entanglement

N-dimensional alternate coined quantum walks from a dispersion relation perspective

“…Indeed a careful analysis reveals that the fine structure of the probability distribution in Fig. 1(c) is very similar to that appearing in conical refraction (the so-called Pogendorf rings [18]), a result to be reported elsewhere with full mathematical details [21]. It is interesting to compare this dynamics with that shown in Fig.…”

“…[19] for QW (1) , i.e., in terms of known solutions of simple linear wave equations. One can then envisage more or less straightforward generalizations of the results there discussed to the two-dimensional case (we study this elsewhere for Grover-QW (2) [21]). But the existence of DPs is particularly appealing and constitutes a qualitative difference with the one-dimensional case.…”

“…The dispersion relation for Grover-QW (2) consists of four sheets, because the coin space is four-dimensional, and can be found in Refs. [14,21]. In our notation, they read ω 1,2 = 0, π and ω 3,4 = ± arccos [(cos u + cos v) /2].…”