Deterministic all-versus-nothing proofs of bell nonlocality induced from qudit non-stabilizer states

Recently, a type of deterministic all-versus-nothing proofs of Bell nonlocality, induced from qubit non-stabilizer states, was proposed, diverging from the tradition where such proofs are typically derived from stabilizer states. Moreover, through the application of a specific map, one can derive certain trivial (dimensionally reducible) versions of such proofs for qudits (d d being even). Nevertheless, it remains unknown whether high-dimensional non-stabilizer states can induce nontrivial deterministic all-versus-nothing proofs of Bell nonlocality. In this study, we provide an example induced from a specific four-qudit non-stabilizer state (with d = 4), demonstrating the feasibility of constructing such proofs in high-dimensional scenarios.