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Quantum systems of high dimensions are attracting a lot of attention becausethey feature interesting properties when it comes to observing entanglement orother forms of correlations. In particular, their improved resistance to noiseis favourable for experiments in quantum communication or quantum cryptography.However, witnessing this high-dimensional nature remains challenging,especially when the assumptions on the parties involved are weak, typicallywhen one of them is considered as a black box. In this context, the concept ofgenuine high-dimensional steering has been recently introduced andexperimentally demonstrated [Phys. Rev. Lett. 126, 200404 (2021)]; it allowsfor a one-sided device-independent certification of the dimension of abipartite shared state by only using two measurements. Here I overcome thislimitation by developing, for more than two measurements, universal bounds onthe incompatibility robustness, turned into meaningful dimension certificates.Interestingly, even though the resulting bounds are quite loose, they stilloften offer an increased resistance to noise and could then be advantageouslyemployed in experiments.

Robust genuine high-dimensional steering with many measurements