  In a well-known result [Werner2001], Werner classified all tight quantum
teleportation and dense coding schemes, showing that they correspond to unitary
error bases. Here tightness is a certain dimensional restriction: the quantum
system to be teleported and the entangled resource must be of dimension d, and
the measurement must have d^2 outcomes.
  In this work we generalise this classification so as to remove the
dimensional restriction altogether, thereby resolving an open problem raised in
that work. In fact, we classify not just teleportation and dense coding
schemes, but entanglement-reversible channels. These are channels between
finite-dimensional C*-algebras which are reversible with the aid of an
entangled resource state, generalising ordinary reversibility of a channel.
  In Werner's classification, a bijective correspondence between tight
teleportation and dense coding schemes was shown: swapping Alice and Bob's
operations turns a teleportation scheme into a dense coding scheme and vice
versa. We observe that this property generalises ordinary invertibility of a
channel; we call it entanglement-invertibility. We show that
entanglement-invertible channels are precisely the quantum bijections
previously studied in the setting of quantum combinatorics [Musto2018], which
are classified in terms of the representation theory of the quantum permutation
group.
