  We study maximal coin-position entanglement generation via a discrete-time
quantum walk, in which the coin operation is randomly selected from one of two
coin operators set at each step. We solve maximal entanglement generation as an
optimization problem with quantum process fidelity as the cost function. Then
we determine the maximal entanglement that can be rigorously generated for any
step beyond the second regardless of initial condition with appropriate coin
sequences. The simplest coin sequence comprising Hadamard and identity
operations is equivalent to the generalized elephant quantum walk, which
exhibits an increasingly faster spreading in terms of probability distribution.
Experimentally, we demonstrate a ten-step quantum walk driven by such coin
sequences with linear optics, and thereby show the desired high-dimensional
bipartite entanglement as well as the transport behavior of faster spreading.
