of concatenation. Similarly, m invocations of the channel corresponds to . The quantity is therefore a measure of the ability of the channel to transmit words of length n faithfully by being invoked m times. This leads to the following definition: A non-negative real number r is an achievable rate of with respect to if For all sequences where and , we have A sequence can be viewed as representing a message consisting of possibly infinite number of words. The limit supremum condition in the definition says that, in the limit, faithful transmission can be achieved by invoking the channel no more than r times the length of a word. One can also say that r is the number of letters per invocation of the channel that can be sent without error. The channel capacity of with respect to , denoted by is the supremum of all achievable rates. From the definition, it is vacuously true that 0 is an achievable rate for any channel. Important examples As stated before, for a system with observable algebra , the ideal channel is by definition the identity map . Thus for a purely n dimensional quantum system, the ideal channel is the identity map on the space of n × n matrices . As a slight abuse of notation, this ideal quantum channel will be also denoted by . Similarly, a classical system with output algebra will have an ideal channel denoted by the same symbol. We can now state some fundamental channel capacities. The channel capacity of the classical ideal channel with respect to a quantum ideal channel is This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel. Moreover, the following equalities hold: The above says, for instance, an ideal quantum channel is no more efficient at transmitting classical information than an ideal classical channel. When n = m, the best one can achieve is one bit per qubit. It is relevant to note here that both of the above bounds on capacities can be broken, with the aid of entanglement. The entanglement-assisted teleportation scheme allows one to transmit quantum information using a classical channel. Superdense coding. achieves two bit per qubit. These results indicate the significant role played by entanglement in quantum communication. Classical and quantum channel capacities Using the same notation as the previous subsection, the classical capacity of a