  We investigate the problem of teleporting an unknown qubit state to a
recipient via a channel of $2\L$ qubits. In this procedure a protocol is
employed whereby $\L$ Bell state measurements are made and information based on
these measurements is sent via a classical channel to the recipient. Upon
receiving this information the recipient determines a local gate which is used
to recover the original state. We find that the $2^{2\L}$-dimensional Hilbert
space of states available for the channel admits a decomposition into four
subspaces. Every state within a given subspace is a perfect channel, and each
sequence of Bell measurements projects $2\L$ qubits of the system into one of
the four subspaces. As a result, only two bits of classical information need be
sent to the recipient for them to determine the gate. We note some connections
between these four subspaces and ground states of many-body Hamiltonian
systems, and discuss the implications of these results towards understanding
entanglement in multi-qubit systems.
