  High-dimensional quantum steering can be seen as a test for the
dimensionality of entanglement, where the devices at one side are not
characterized. As such, it is an important component in quantum informational
protocols that make use of high-dimensional entanglement. Although it has been
recently observed experimentally, the phenomenon of high-dimensional steering
is lacking a general certification procedure. We provide necessary and
sufficient conditions to certify the entanglement dimension in a steering
scenario. These conditions are stated in terms of a hierarchy of semidefinite
programs, which can also be used to quantify the phenomenon using the steering
dimension robustness. To demonstrate the practical viability of our method, we
characterize the dimensionality of entanglement in steering scenarios prepared
with maximally entangled states measured in mutually unbiased bases. Our
methods give significantly stronger bounds on the noise robustness necessary to
experimentally certify high-dimensional entanglement.
