Editing Quantum channel (section)

=== Schrödinger picture ===

Consider quantum channels that transmit only quantum information.  This is precisely a [[quantum operation]], whose properties we now summarize.

Let <math>H_A</math> and <math>H_B</math> be the state spaces (finite-dimensional [[Hilbert space]]s) of the sending and receiving ends, respectively, of a channel. <math>L(H_A)</math> will denote the family of operators on <math>H_A.</math> In the [[Schrödinger picture]], a purely quantum channel is a map <math> \Phi</math> between [[density matrix|density matrices]] acting on <math>H_A</math> and <math>H_B</math> with the following properties:{{sfn|Wilde|2017|at=§4.4.1}}

#As required by postulates of quantum mechanics, <math> \Phi</math> needs to be linear.
#Since density matrices are positive, <math> \Phi</math> must preserve the [[cone (linear algebra)|cone]] of positive elements.  In other words, <math> \Phi</math> is a [[Choi's theorem on completely positive maps|positive map]].
#If an [[ancilla (quantum computing)|ancilla]] of arbitrary finite dimension ''n'' is coupled to the system, then the induced map <math>I_n \otimes \Phi,</math> where ''I''<sub>''n''</sub> is the identity map on the ancilla, must also be positive. Therefore, it is required that <math>I_n \otimes \Phi</math> is positive for all ''n''. Such maps  are called [[completely positive]].
#Density matrices are specified to have trace 1, so <math> \Phi</math> has to preserve the trace.

The adjectives '''completely positive and trace preserving''' used to describe a map are sometimes abbreviated '''CPTP'''.  In the literature, sometimes the fourth property is weakened so that <math> \Phi</math> is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP.