  Dense coding is arguably the protocol that launched the field of quantum
communication. Today, however, more than a decade after its initial
experimental realization, the channel capacity remains fundamentally limited as
conceived for photons using linear elements. Bob can only send to Alice three
of four potential messages owing to the impossibility of carrying out the
deterministic discrimination of all four Bell states with linear optics,
reducing the attainable channel capacity from 2 to log_2 3 \approx 1.585 bits.
However, entanglement in an extra degree of freedom enables the complete and
deterministic discrimination of all Bell states. Using pairs of photons
simultaneously entangled in spin and orbital angular momentum, we demonstrate
the quantum advantage of the ancillary entanglement. In particular, we describe
a dense-coding experiment with the largest reported channel capacity and, to
our knowledge, the first to break the conventional linear-optics threshold. Our
encoding is suited for quantum communication without alignment and satellite
communication.
