Editing Quantum channel (section)

== Examples ==

=== Time evolution ===

For a purely quantum system, the time evolution, at certain time ''t'', is given by

:<math>\rho \rightarrow U \rho \;U^*,</math>

where <math>U = e^{-iH t/\hbar}</math> and ''H'' is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and ''t'' is the time.  This gives a CPTP map in the Schrödinger picture and is therefore a channel.{{sfn|Wilde|2017|at=§4.6.3}} The dual map in the Heisenberg picture is

:<math>A \rightarrow U^* A U.</math>

=== Restriction ===

Consider a composite quantum system with state space <math>H_A \otimes H_B.</math> For a state

:<math>\rho \in H_A \otimes H_B,</math>

the reduced state of ''ρ'' on system ''A'', ''ρ''<sup>''A''</sup>, is obtained by taking the [[partial trace]] of ''ρ'' with respect to the ''B'' system:

:<math> \rho ^A = \operatorname{Tr}_B \; \rho.</math>

The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture.{{sfn|Wilde|2017|at=§4.6.2}} In the Heisenberg picture, the dual map of this channel is

:<math> A \rightarrow A \otimes I_B,</math>

where ''A'' is an observable of system ''A''.

=== Observable ===

An observable associates a numerical value <math>f_i \in \mathbb{C}</math> to a quantum mechanical ''effect'' <math>F_i</math>. <math>F_i</math>'s are assumed to be positive operators acting on appropriate state space and <math display="inline">\sum_i F_i = I</math>. (Such a collection is called a [[POVM]].<ref>{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |year=1993 |publisher=[[Kluwer]] |isbn=0-7923-2549-4 |page=283}}</ref>{{sfn|Bengtsson|Życzkowski|2017|p=271}}) In the Heisenberg picture, the corresponding ''observable map'' <math>\Psi</math> maps a classical observable

:<math>f = \begin{bmatrix} f_1 \\ \vdots \\ f_n \end{bmatrix} \in C(X)</math>

to the quantum mechanical one

:<math>\; \Psi (f) = \sum_i f_i F_i.</math>

In other words, one [[Naimark's dilation theorem|integrates ''f'' against the POVM]] to obtain the quantum mechanical observable. It can be easily checked that <math>\Psi</math> is CP and unital.

The corresponding Schrödinger map <math>\Psi^*</math> takes density matrices to classical states:{{sfn|Wilde|2017|at=§4.6.6}}

:<math>
\Psi (\rho) = \begin{bmatrix} \langle F_1, \rho  \rangle \\ \vdots \\ \langle F_n, \rho \rangle \end{bmatrix}, 
</math>

where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized [[density matrix#C*-algebraic formulation of states|functionals]], and invoking the [[Riesz representation theorem]], we can put

:<math>
\Psi (\rho) = \begin{bmatrix} \rho (F_1) \\ \vdots \\ \rho (F_n) \end{bmatrix}.
</math>

=== Instrument ===

The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called a [[quantum instrument]]. Let <math>\{ F_1, \dots, F_n \}</math> be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map <math>\Phi</math> with pure quantum input <math>\rho \in L(H)</math> and with output space <math>C(X) \otimes L(H)</math>:

:<math>
\Phi (\rho) =  \begin{bmatrix} \rho(F_1) \cdot F_1 \\ \vdots \\ \rho(F_n) \cdot F_n \end{bmatrix}.
</math>

Let

:<math>
f =  \begin{bmatrix} f_1 \\ \vdots \\ f_n \end{bmatrix} \in C(X).
</math>

The dual map in the Heisenberg picture is

:<math>
\Psi (f \otimes A) =  \begin{bmatrix} f_1 \Psi_1(A) \\ \vdots \\ f_n \Psi_n(A)\end{bmatrix}
</math>

where <math>\Psi_i</math> is defined in the following way: Factor <math>F_i = M_i ^2</math> (this can always be done since elements of a POVM are positive) then <math>\; \Psi_i (A) = M_i A M_i</math>.
We see that <math>\Psi</math> is CP and unital.

Notice that <math>\Psi (f \otimes I)</math> gives precisely the observable map. The map

:<math>{\tilde \Psi}(A)= \sum_i \Psi_i (A) = \sum _i M_i A M_i</math>

describes the overall state change.

=== Measure-and-prepare channel ===

Suppose two parties ''A'' and ''B'' wish to communicate in the following manner: ''A'' performs the measurement of an observable and communicates the measurement outcome to ''B'' classically. According to the message he receives, ''B'' prepares his (quantum) system in a specific state. In the Schrödinger picture, the first part of the channel <math> \Phi</math><sub>1</sub> simply consists of ''A'' making a measurement, i.e. it is the observable map:

:<math>\; \Phi_1 (\rho) = \begin{bmatrix} \rho(F_1) \\ \vdots \\ \rho(F_n)\end{bmatrix}.</math>

If, in the event of the ''i''-th measurement outcome, ''B'' prepares his system in state ''R<sub>i</sub>'', the second part of the channel <math> \Phi</math><sub>2</sub> takes the above classical state to the density matrix

:<math>
\Phi_2 \left(\begin{bmatrix} \rho(F_1) \\ \vdots \\ \rho(F_n)\end{bmatrix}\right) = \sum _i \rho (F_i) R_i.
</math>

The total operation is the composition

:<math>\Phi (\rho)= \Phi_2 \circ \Phi_1 (\rho) = \sum _i \rho (F_i) R_i.</math>

Channels of this form are called ''measure-and-prepare'' or ''entanglement-breaking.''<ref>{{cite journal|last=Ruskai |first=Mary Beth |author-link=Mary Beth Ruskai |title=Qubit entanglement breaking channels |journal=Reviews in Mathematical Physics |volume=15 |number=6 |year=2003 |pages=643–662 |doi=10.1142/S0129055X03001710 |arxiv=quant-ph/0302032|bibcode=2003RvMaP..15..643R }}</ref>{{sfn|Wilde|2017|at=§4.6.7}}<ref>{{cite journal|last1=DeBrota |first1=John B. |first2=Blake C. |last2=Stacey |title=Lüders channels and the existence of symmetric-informationally-complete measurements |journal=Physical Review A |volume=100 |number=6 |year=2019 |page=062327 |doi=10.1103/PhysRevA.100.062327 |arxiv=1907.10999|bibcode=2019PhRvA.100f2327D }}</ref><ref>{{cite journal|first1=Satish K. |last1=Pandey |first2=Vern I. |last2=Paulsen |first3=Jitendra |last3=Prakash |first4=Mizanur |last4=Rahaman |title=Entanglement Breaking Rank and the existence of SIC POVMs |arxiv=1805.04583 |journal=Journal of Mathematical Physics |volume=61 |page=042203 |year=2020 |issue=4 |doi=10.1063/1.5045184|bibcode=2020JMP....61d2203P }}</ref>

In the Heisenberg picture, the dual map <math>\Phi^* = \Phi_1^* \circ \Phi_2 ^*</math> is defined by

:<math>\; \Phi^* (A) = \sum_i R_i(A) F_i.</math>

A measure-and-prepare channel can not be the identity map. This is precisely the statement of the [[no teleportation theorem]], which says classical teleportation (not to be confused with [[quantum teleportation|entanglement-assisted teleportation]]) is impossible. In other words, a quantum state can not be measured reliably.

In the [[channel-state duality]], a channel is measure-and-prepare if and only if the corresponding state is [[separable state|separable]]. Actually, all the states that result from the partial action of a measure-and-prepare channel are separable, which is why measure-and-prepare channels are also known as entanglement-breaking channels.

=== 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.

=== Teleportation ===

In [[quantum teleportation]], a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel.  The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver.  Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle.  This measurement results in classical information that must be sent to the receiver to complete the teleportation.  Importantly, the classical information can be sent after the quantum channel has ceased to exist.