Editing Quantum channel (section)

=== 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.