Editing Quantum channel (section)

== Memoryless quantum channel ==

We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional.

The '''memoryless''' in the section title carries the same meaning as in classical [[information theory]]: the output of a channel at a given time depends only upon the corresponding input and not any previous ones.

=== 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.

=== Heisenberg picture ===

Density matrices acting on ''H<sub>A</sub>'' only constitute a proper subset of the operators on ''H<sub>A</sub>'' and same can be said for system ''B''.  However, once a linear map <math> \Phi</math> between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend <math> \Phi</math> uniquely to the full space of operators. This leads to the adjoint map <math> \Phi^*</math>, which describes the action of <math> \Phi</math> in the [[Heisenberg picture]]:{{sfn|Wilde|2017|at=§4.4.5}}

The spaces of operators ''L''(''H''<sub>''A''</sub>) and ''L''(''H''<sub>''B''</sub>) are Hilbert spaces with the [[Hilbert–Schmidt operator|Hilbert–Schmidt]] inner product. Therefore, viewing <math>\Phi : L(H_A) \rightarrow L(H_B)</math> as a map between Hilbert spaces, we obtain its adjoint <math> \Phi</math><sup>*</sup> given by

:<math>\langle A , \Phi(\rho) \rangle = \langle \Phi^*(A) , \rho \rangle .</math>

While <math> \Phi</math> takes states on ''A'' to those on ''B'', <math> \Phi^*</math> maps observables on system ''B'' to observables on ''A''. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.

It can be directly checked that if <math> \Phi</math> is assumed to be trace preserving, <math> \Phi^*</math> is [[unital map|unital]], that is,<math> \Phi^*(I) = I</math>. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.

=== Classical information ===

So far we have only defined a quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators:

:<math>\Psi : L(H_B) \rightarrow L(H_A)</math>

that is unital and completely positive ('''CP'''). The operator spaces can be viewed as finite-dimensional [[C*-algebra]]s. Therefore, we can say a channel is a unital CP map between C*-algebras:

:<math>\Psi : \mathcal{B} \rightarrow \mathcal{A}.</math>

Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions <math>C(X)</math> on some set <math>X</math>. We assume <math>X</math> is finite so <math>C(X)</math> can be identified with the ''n''-dimensional Euclidean space <math>\mathbb{R}^n</math> with entry-wise multiplication.

Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define <math>\mathcal{B}</math> to include the relevant classical observables. An example of this would be a channel

:<math>\Psi : L(H_B) \otimes C(X) \rightarrow L(H_A).</math>

Notice <math>L(H_B) \otimes C(X)</math> is still a C*-algebra. An element <math>a</math> of a C*-algebra <math>\mathcal{A}</math> is called positive if <math>a = x^{*} x</math> for some <math>x</math>. Positivity of a map is defined accordingly.  This characterization is not universally accepted; the [[quantum instrument]] is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a [[Frobenius algebra]] or [[Frobenius category]].