Editing Quantum channel (section)

== Channel capacity ==

=== The cb-norm of a channel ===

Before giving the definition of channel capacity, the preliminary notion of the '''norm of complete boundedness''', or '''cb-norm''' of a channel needs to be discussed. When considering the capacity of a channel <math>\Phi</math>, we need to compare it with an "ideal channel" <math>\Lambda</math> . For instance, when the input and output algebras are identical, we can choose <math>\Lambda</math> to be the identity map. Such a comparison requires a [[metric (mathematics)|metric]] between channels.
Since a channel can be viewed as a linear operator, it is tempting to use the natural [[operator norm]]. In other words, the closeness of <math>\Phi</math> to the ideal channel <math>\Lambda</math> can be defined by

:<math>\| \Phi - \Lambda \| = \sup \{ \| (\Phi - \Lambda)(A)\|  \;|\;  \|A\| \leq 1 \}.</math>

However, the operator norm may increase when we tensor <math>\Phi</math> with the identity map on some ancilla.

To make the operator norm even a more undesirable candidate, the quantity

:<math>\| \Phi \otimes I_n \|</math>

may increase without bound as <math>n \rightarrow \infty.</math> The solution is to introduce, for any linear map <math>\Phi</math> between C*-algebras, the cb-norm

:<math>\| \Phi \|_{cb} = \sup _n \| \Phi \otimes I_n \|.</math>

=== Definition of channel capacity ===

The mathematical model of a channel used here is same as the [[channel capacity|classical one]].

Let <math>\Psi :\mathcal{B}_1 \rightarrow \mathcal{A}_1</math> be a channel in the Heisenberg picture and <math>\Psi_{id} : \mathcal{B}_2 \rightarrow \mathcal{A}_2</math> be a chosen ideal channel. To make the comparison possible, one needs to encode and decode Φ via appropriate devices, i.e. we consider the composition

:<math>{\hat \Psi} = D \circ \Phi \circ E : \mathcal{B}_2 \rightarrow \mathcal{A}_2 </math>

where ''E'' is an encoder and ''D'' is a decoder. In this context, ''E'' and ''D'' are unital CP maps with appropriate domains. The quantity of interest is the ''best case scenario'':

:<math>\Delta ({\hat \Psi}, \Psi_{id}) = \inf_{E,D} \| {\hat \Psi} - \Psi_{id} \|_{cb}</math>

with the infimum being taken over all possible encoders and decoders.

To transmit words of length ''n'', the ideal channel is to be applied ''n'' times, so we consider the tensor power

:<math>\Psi_{id}^{\otimes n} = \Psi_{id} \otimes \cdots \otimes \Psi_{id}.</math>

The <math>\otimes</math> operation describes ''n'' inputs undergoing the operation <math>\Psi_{id}</math> independently and is the quantum mechanical counterpart of [[concatenation]]. Similarly, ''m invocations of the channel'' corresponds to <math>{\hat \Psi} ^{\otimes m}</math>.

The quantity

:<math>\Delta ( {\hat \Psi}^{\otimes m}, \Psi_{id}^{\otimes n} )</math>

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 <math>\Psi</math> with respect to <math>\Psi_{id}</math>''' if

:For all sequences <math>\{ n_{\alpha} \}, \{ m_{\alpha} \} \subset \mathbb{N}</math> where <math>m_{\alpha}\rightarrow \infty</math> and <math>\lim \sup _{\alpha} (n_{\alpha}/m_{\alpha}) < r</math>, we have

:<math>\lim_{\alpha} \Delta ( {\hat \Psi}^{\otimes m_{\alpha}}, \Psi_{id}^{\otimes n_{\alpha}} ) = 0.</math>

A sequence <math>\{ n_{\alpha} \}</math> 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 <math>\Psi</math> with respect to <math>\Psi_{id}</math>''', denoted by <math>\;C(\Psi, \Psi_{id})</math> 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 <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.

=== Classical and quantum channel capacities ===

Using the same notation as the previous subsection, the '''classical capacity''' of a channel Ψ is

:<math>C(\Psi, \mathbb{C}^2),</math>

that is, it is the capacity of Ψ with respect to the ideal channel on the classical one-bit system <math>\mathbb{C}^2</math>.

Similarly the '''quantum capacity''' of Ψ is

:<math>C(\Psi, \mathbb{C}^{2 \times 2}),</math>

where the reference system is now the one qubit system <math>\mathbb{C}^{2 \times 2}</math>.