This paper shows a way to test whether a physical theory really is ordinary finite‑dimensional quantum theory. The authors construct a finite collection of two‑body correlations (probabilities for outcomes on pairs of systems) with two conditions. First, a theory must reproduce those correlations. Second, the way those correlations behave must be stable when systems are sent through a sequence of quantum teleportation steps. The claim is that the only probabilistic theory that meets both conditions is finite‑dimensional quantum mechanics (QM). In that sense the paper gives an operational, experiment‑style certificate for QM.

Concretely, the authors build experiments based on entanglement swapping and teleportation. For the simplest case (a qubit) they describe a three‑party entanglement swapping test with Alice, Bob and Erwin. Alice and Bob each have three settings with two outcomes. Erwin has three settings with four outcomes. Alice–Erwin and Erwin–Bob share Bell states, and Erwin performs Bell‑basis measurements or rotated versions of them. The correlations between Alice and Bob, conditioned on Erwin’s outcomes, are chosen so that they encode the symmetry generators of the qubit state space.

At a high level the idea is this. Teleportation is a protocol that moves quantum states using entanglement and measurements. If the observed two‑body correlations survive arbitrary repetitions of teleportation, then those correlations must come from dynamical maps that generate a group of transformations. Under the authors’ assumptions those generators often give the full unitary group on the system. The second condition is therefore a hierarchy of tests: one can check stability under N successive teleportation steps for increasing N. For systems in (C^d)^{⊗ n} the first condition (realizing the target correlations) can be checked with a number of measurement settings that grows only polynomially in d and n, so the construction is finite and scalable in that precise sense.