  We experimentally investigate the properties of hyperentangled states
displaying simultaneous entanglement in multiple degrees of freedom, and find
that Bell tests beyond the standard Clauser, Horne, Shimony, Holt inequality
can reveal a higher-dimensional nature in a device-independent way.
Specifically, we show that hyperentangled states possess more than just
simultaneous entanglement in separate degrees of freedom but also entanglement
in a higher dimensional Hilbert space. We also verify the steerability of
hyperentangled quantum states by steering different photonic degrees of
freedom.
