Editing Quantum channel (section)

=== Important examples ===

As stated before, for a system with observable algebra <math>\mathcal{B}</math>, the ideal channel <math>\Psi_{id}</math> is by definition the identity map <math>I_{\mathcal{B}}</math>. Thus for a purely ''n'' dimensional quantum system, the ideal channel is the identity map on the space of ''n''&nbsp;×&nbsp;''n'' matrices <math>\mathbb{C}^{n \times n}</math>. As a slight abuse of notation, this ideal quantum channel will be also denoted by <math>\mathbb{C}^{n \times n}</math>. Similarly, a classical system with output algebra <math>\mathbb{C}^m</math> 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 <math>\mathbb{C}^m</math> with respect to a quantum ideal channel <math>\mathbb{C}^{n \times n}</math> is

:<math>C(\mathbb{C}^m, \mathbb{C}^{n \times n}) = 0.</math>

This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel.

Moreover, the following equalities hold:

:<math>
C(\mathbb{C}^m, \mathbb{C}^n) = C(\mathbb{C}^{m \times m}, \mathbb{C}^{n \times n}) 
= C( \mathbb{C}^{m \times m}, \mathbb{C}^{n} ) = \frac{\log n}{\log m}.
</math>

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 [[quantum entanglement|entanglement]]. The [[quantum teleportation|entanglement-assisted teleportation scheme]] allows one to transmit quantum information using a classical channel. [[Superdense coding]] achieves two bits per qubit. These results indicate the significant role played by entanglement in quantum communication.