  Entanglement lies at the heart of quantum mechanics $-$ as a fundamental tool
for testing its deep rift with classical physics, while also providing a key
resource for quantum technologies such as quantum computation and cryptography.
In 1987 Greenberger, Horne, and Zeilinger realized that the entanglement of
more than two particles implies a non-statistical conflict between local
realism and quantum mechanics. The resulting predictions were experimentally
confirmed by entangling three photons in their polarization. Experimental
efforts since have singularly focused on increasing the number of particles
entangled, while remaining in a two-dimensional space for each particle. Here
we show the experimental generation of the first multi-photon entangled state
where both $-$ the number of particles and the number of dimensions $-$ are
greater than two. Interestingly, our state exhibits an asymmetric entanglement
structure that is only possible when one considers multi-particle entangled
states in high dimensions. Two photons in our state reside in a
three-dimensional space, while the third lives in two dimensions. Our method
relies on combining two pairs of photons, high-dimensionally entangled in their
orbital angular momentum, in such a way that information about their origin is
erased. Additionally, we show how this state enables a new type of "layered"
quantum cryptographic protocol where two parties share an additional layer of
secure information over that already shared by all three parties. In addition
to their application in novel quantum communication protocols, such asymmetric
entangled states serve as a manifestation of the complex dance of correlations
that can exist within quantum mechanics.
