Editing Quantum channel (section)

=== Pure channel ===

Consider the case of a purely quantum channel <math>\Psi</math> in the Heisenberg picture. With the assumption that everything is finite-dimensional, <math>\Psi</math> is a unital CP map between spaces of matrices

:<math>\Psi : \mathbb{C}^{n \times n} \rightarrow \mathbb{C}^{m \times m}.</math>

By [[Choi's theorem on completely positive maps]], <math>\Psi</math> must take the form

:<math>\Psi (A) = \sum_{i = 1}^N K_i A K_i^*</math>

where ''N'' ≤ ''nm''. The matrices ''K''<sub>''i''</sub> are called '''[[Kraus operator]]s''' of <math>\Psi</math> (after the German physicist [[Karl Kraus (physicist)|Karl Kraus]], who introduced them).<ref>{{Cite book| publisher = Springer-Verlag| isbn = 978-3-5401-2732-1| last = Kraus| first = Karl| author-link = Karl Kraus (physicist)| title = States, effects, and operations: fundamental notions of quantum theory| series = Lectures in mathematical physics at the University of Texas at Austin| volume = 190| date = 1983
| url = https://books.google.com/books?id=fRBBAQAAIAAJ}}</ref><ref>{{Cite journal|last1=Barnum|first1=Howard|last2=Nielsen|first2=M. A.|author-link2=Michael Nielsen|last3=Schumacher|first3=Benjamin|author-link3=Benjamin Schumacher|date=1 June 1998|title=Information transmission through a noisy quantum channel|journal=[[Physical Review A]]|language=en|volume=57|issue=6|pages=4153–4175|arxiv=quant-ph/9702049|doi=10.1103/PhysRevA.57.4153|bibcode=1998PhRvA..57.4153B}}</ref><ref>{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Jacobs|first2=Kurt|date=16 May 2001|title=Information-tradeoff relations for finite-strength quantum measurements|journal=[[Physical Review A]]|language=en|volume=63|issue=6|pages=062305|arxiv=quant-ph/0009101|bibcode=2001PhRvA..63f2305F|doi=10.1103/PhysRevA.63.062305}}</ref> The minimum number of Kraus operators is called the Kraus rank of <math>\Psi</math>. A channel with Kraus rank 1 is called '''pure'''. The time evolution is one example of a pure channel. This terminology again comes from the channel-state duality. A channel is pure if and only if its dual state is a pure state.