SECTION 1. Introduction High-dimensional quantum states and entanglement underpin many of the advantages promised by quantum information processors. The information encoding in high-dimensional systems enhances channel capacities in quantum communication, security and noise resistance in quantum cryptography, and performances in quantum error correction and fault-tolerant quantum computation. These promising advantages motivate additional efforts to¬wards the formulation of effective protocols for the generation, manipulation, and certification of such a quantum resource [1]. Despite significant experimental advances, the implementation of these tasks remains demanding, especially due to the arduousness of controlling operations in large Hilbert spaces. In this work we illustrate an approach to generate high-dimensional entangled states based on discrete-time quantum walks (QWs).

SECTION 2. Quantum Walks and Qudit Engineering In discrete-time quantum walks a particle leaving on a lattice of d sites is equipped with an internal two¬dimensional degree of freedom q = {| ↑⟩, | ↓⟩}. The evolution is described by two operators, the coin C that operates in the two-dimensional subsystem and the shift S, that moves the particle according to the state of the coin: S=∑d|d+1⟩⟨d|⊗|↑⟩⟨↑|+|d−1⟩⟨d|⊗|↓⟩⟨↓|(1)

Given this very general formulation, QWs have found experimental realizations in various physical systems and in particular in photonic platforms [2]. In Fig. 1a we illustrate an implementation in the angular momentum of single-photon states. The orbital angular momentum (OAM) encodes the position on the lattice and the spin angular momentum (SAM), i.e the circular polarization, the coin [3]. The output state produced by such QW displays correlation between these two degrees of freedom. Then the coin operations can be engineered by proper numerical optimization, to generate agiven target state in the position ofthe particle [4]. In this way QWs produce high-dimensional states encoded in the OAM [5], which are extensively employed in quantum communication and cryptography.

SECTION 3. High-dimensional Entanglement Transfer In the previous section we have illustrated how to exploit the position-coin correlations in QW to engineer high-dimensional state. Here we generalize the concept to the two walkers case which share entanglement in the coin. This scheme basically leverages the interface between systems of different dimensions, to realize an effective entanglement transfer between a low- and a high-dimensional degree of freedom [6]. Under particular conditions two local QW routines and proper local coin projections allow to transfer the initial entanglement between the two coins to the walkers position degrees of freedom. The high-dimensional Hilbert space of the walkers positions is then used to accumulate entanglement by iterating the transfer protocol. At each iteration a Bell-like state is generated and such amount of entanglement is transferred by suitable QWs and coin measurements. Therefore the two local QW routines are the mediators to transfer entanglement and generate high-dimensional entangled states. This scheme offers a promising two-way interface to reliably transfer and retrieve entanglement between different information carriers. Furthermore, being based on QW dynamics, it is very general and can be employed in different experimental frameworks and platforms. Here we propose the same encoding in the angular momentum of single photon states illustrated in the previous section and in the caption of Fig 1.