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Quantum channel
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=== 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 Ξ¦ 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.
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