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\section*{quantum operation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InTermsOfMatrices}{In terms of matrices}\dotfill \pageref*{InTermsOfMatrices} \linebreak
\noindent\hyperlink{InTermsOfCompactClosedCategories}{In terms of compact closed categories}\dotfill \pageref*{InTermsOfCompactClosedCategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_complete_positivity}{Characterization of complete positivity}\dotfill \pageref*{characterization_of_complete_positivity} \linebreak
\noindent\hyperlink{universal_property}{Universal property}\dotfill \pageref*{universal_property} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{quantum_measurement_and_povms}{Quantum measurement and POVMs}\dotfill \pageref*{quantum_measurement_and_povms} \linebreak
\noindent\hyperlink{systems_in_a_bath}{Systems in a bath}\dotfill \pageref*{systems_in_a_bath} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Any physical process is supposed to take [[physical states]] into physical states ([[Schrödinger picture]]). If [[density matrices]] are used to describe [[quantum states]] in [[quantum mechanics]], then it must be some operation that sends density matrices to density matrices. So for finite-dimensional state spaces a process should be a [[linear map]] of [[vector spaces]] of [[matrices]]

\begin{displaymath}
U : Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})
\end{displaymath}
(so far this is a general ``[[superoperator]]'') that preserves the subset of [[density matrices]], in that

\begin{itemize}%
\item it preserves the [[trace]] of matrices;


\item takes hermitian matrices with non-negative eigenvalues to hermitian matrices with non-negative eigenvalues.



\end{itemize}
Such a map is then called a \emph{quantum operation}. The notion of a quantum operation is built from the [[Stinespring factorization theorem]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We first give the traditional definition in terms of [[linear algebra]] and [[matrices]] in

\begin{itemize}%
\item \hyperlink{InTermsOfMatrices}{In terms of matrices}

\end{itemize}
Then we consider the general abstract formulation

\begin{itemize}%
\item \hyperlink{InTermsOfCompactClosedCategories}{In terms of compact closed categories}

\end{itemize}
\hypertarget{InTermsOfMatrices}{}\subsubsection*{{In terms of matrices}}\label{InTermsOfMatrices}

Let $k,n \in \mathbb{N}$.

A matrix $A \in Mat(n \times n, \mathbb{C})$ is called \emph{positive} if it is hermitian -- if $A^\dagger = A$ -- and if all its eigenvalues (which then are necessarily real) are non-negative.

A linear map (morphism of [[vector space]]s of matrices)

\begin{displaymath}
\Phi : Mat(n \times n, \mathbb{C}) \to 
   Mat(k \times k, \mathbb{C})
\end{displaymath}
is called \emph{positive} if it takes positive matrices to positive matrices.

The map $\Phi$ is called \emph{completely positive} if for all $p \in \mathbb{N}$ the [[tensor product]]

\begin{displaymath}
\Phi \otimes Id_{Mat(p\times p),\mathbb{C}} :
  Mat(n \times n , \mathbb{C})\otimes
  Mat(p \times p , \mathbb{C})
  \to
  Mat(k \times k , \mathbb{C})\otimes
  Mat(p \times p , \mathbb{C})
\end{displaymath}
is positive.

A \emph{quantum operation} (or quantum channel) is a map that is both completely positive and trace preserving (often abbreviated to CPTP).

\hypertarget{InTermsOfCompactClosedCategories}{}\subsubsection*{{In terms of compact closed categories}}\label{InTermsOfCompactClosedCategories}

\ldots{} due to (\hyperlink{Selinger}{Selinger 05}) \ldots{} see for instance (\hyperlink{CoeckeHeunen11}{Coecke-Heunen 11, section 2}) for a quick summary \ldots{}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{characterization_of_complete_positivity}{}\subsubsection*{{Characterization of complete positivity}}\label{characterization_of_complete_positivity}

\begin{theorem}
\label{}\hypertarget{}{}
A map $\Phi$ as above is \emph{completely positive} precisely if there exists a [[set]] $I$ and an $I$-family $\{E_i \in Mat(k \times n, \mathbb{C}| i \in I)\}$ of matrices, such that for all $A \in Mat(n \times n, \mathbb{C})$ we have

\begin{displaymath}
\Phi(A) = \sum_{i \in I} E_i A E_i^\dagger
  \,.
\end{displaymath}
Moreover, such $\Phi$ preserves the trace of matrices precisely if

\begin{displaymath}
\sum_{i \in I} E_i^\dagger E_i = Id_{Mat(n \times n, \mathbb{C})}
  \,.
\end{displaymath}
\end{theorem}
This is originally due to (\hyperlink{Stinespring55}{Stinespring 55}). The decomposition in the theorem is called \emph{Kraus decomposition} after (\hyperlink{Kraus71}{Kraus 71}). See also (\hyperlink{Choi76}{Choi 76, theorem 1}). A brief review is for instance in (\hyperlink{Kuperberg05}{Kuperberg 05, theorem 1.5.1}). A general abstract proof in terms of [[dagger category|†-categories]] is given in (\hyperlink{Selinger05}{Selinger 05}). A characterization of completely positive maps entirely in terms of $\dagger$-categories is given in (\hyperlink{Coecke07}{Coecke 07}).

The matrices $\{E_i\}$ that are associated to a completely positive and trace-preserving map by the above theorem are called \textbf{Kraus operators}.

In the physics literature the above theorem is then phrased as: \emph{Every quantum channel can be represented using Kraus operators} .

Notice that the identity map is clearly completely positive and trace preserving, and that the composite of two maps that preserve positivity and trace clearly still preserves positivity and trace. Therefore we obtain a [[category]] $QChan \subset Vect$ -- a [[subcategory]] of [[Vect]]${}_{\mathbb{C}}$ -- whose

\begin{itemize}%
\item objects are the vector spaces $Mat(n \times n, \mathbb{C})$ for all $n \in \mathbb{N}$;


\item morphism are completely positive and trace-preserving linear maps $\Phi : Mat(n\times n , \mathbb{C}) \to Mat(m \times m, \mathbb{C})$;


\item composition of morphisms is, of course, the composition in [[Vect]], i.e. the ordinary composition of linear maps.



\end{itemize}
See also [[extremal quantum channels]] and [[graphical quantum channels]].

\hypertarget{universal_property}{}\subsection*{{Universal property}}\label{universal_property}

The category of natural numbers and quantum operations between them is a [[semicartesian monoidal category]] with the monoidal structure given by multiplication of numbers. Being semicartesian, the monoidal unit (the number $1$) has a unique morphism to it from any object: this morphism is the trace.

In fact, this category has the universal property of the semicartesian reflection of the monoidal category of isometries. This is the category whose objects are natural numbers, considered as Hilbert spaces, and whose morphisms are isometries between them, where an [[isometry]] $m\to n$ is an $m\times n$ complex matrix $V$ such that $VV*=I$. In detail, the universal property says that for any strict semicartesian monoidal category $\mathcal{D}$ and any monoidal functor $\mathbf{Isometries}\to \mathcal{D}$, there is a unique symmetric monoidal functor making the following diagram commute:

\begin{displaymath}
\itexarray{
\mathbf{Isometries} &\rightarrow& \mathbf{Quantum Channels} \\
&\searrow&\downarrow\\
&& \mathcal{D}
}
\end{displaymath}
This fits a physical intuition as follows. Suppose that the isometries are a model of reality, as in the the many worlds interpretation and the Church of the larger Hilbert space. But in practice the observer cannot access the entirety of reality, and so some bits are hidden. The canonical way to model this hiding is to do it freely, which is to form the semicartesian reflection.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{quantum_measurement_and_povms}{}\subsubsection*{{Quantum measurement and POVMs}}\label{quantum_measurement_and_povms}

A [[quantum measurement]] is formally represented by a quantum operation that is induced by a [[positive-operator valued probability measure]] (POVM).

\hypertarget{systems_in_a_bath}{}\subsubsection*{{Systems in a bath}}\label{systems_in_a_bath}

A very common example of this formalism comes from its use in [[open quantum systems]], that is systems that are coupled to an environment. Let $\rho$ be the state of some quantum system and $\rho_{env}$ be the state of the environment. The action of a unitary transformation, $U$, on the system is

\begin{displaymath}
T(\rho) = Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[quantum mechanics in terms of dagger-compact categories]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The Kraus-decomposition characterization of completely positive maps is due to

\begin{itemize}%
\item W. Forrest Stinespring, \emph{Positive functions on $C^\ast$-algebras}, Proc. Amer. Math. Soc. 6 (1955), 211--216.

\end{itemize}
\begin{itemize}%
\item K. Kraus, \emph{General state changes in quantum theory}, Ann. Physics 64 (1971), no. 2, 311--335.

\end{itemize}
\begin{itemize}%
\item M. Choi, \emph{Completely positive linear maps on complex matrices}, Linear Algebra and its Applications Volume 10, Issue 3, (1975), Pages 285-290

\end{itemize}
Reviews and surveys include

\begin{itemize}%
\item [[Greg Kuperberg]], section 1.5 of \emph{A concise introduction to quantum probability, quantum mechanics, and quantum computation}, 2005 (\href{http://www.math.ucdavis.edu/~greg/intro-2005.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Michael Nielsen, Isaac Chuang, section 8.2 of \emph{Quantum Computation and Quantum Information}, Cambridge University Press, Cambridge (2000) ([[NielsenChuangQuantumComputation.pdf:file]])

\end{itemize}
See also

\begin{itemize}%
\item Caleb J. O'Loan (2009), \emph{Topics in Estimation of Quantum Channels} , PhD thesis, University of St. Andrews, (\href{http://arxiv.org/abs/1001.3971}{arXiv})


\item Christian B. Mendl, Michael M. Wolf, \emph{Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem} , Commun. Math. Phys. 289, 1057-1096 (2009) (\href{http://arxiv.org/abs/0806.2820}{arXiv:0806.2820})


\item Smolin, John A., Verstraete, Frank, and Winter, Andreas \emph{Entanglement of assistance and multipartite state distillation} , Phys. Rev. A, vol. 72, 052317, 2005 (\href{http://arxiv.org/abs/quant-ph/0505038}{arXiv:quant-ph/0505038})


\item John Watrous, \emph{Mixing doubly stochastic quantum channels with the completely depolarizing channel} (2008) (\href{http://arxiv.org/abs/0807.2668}{arXiv})



\end{itemize}
The description of completely positive maps in terms of [[dagger-categories]] (see at \emph{[[finite quantum mechanics in terms of dagger-compact categories]]}) goes back to

\begin{itemize}%
\item [[Peter Selinger]], \emph{Dagger-compact closed categories and completely positive maps}, Electronic Notes in Theoretical Computer Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 ([[SelingerPositiveMaps.pdf:file]], \href{http://www.mscs.dal.ca/~selinger/papers/dagger.ps}{ps})

\end{itemize}
\begin{itemize}%
\item [[Bob Coecke]], \emph{Complete positivity without compactness}, 2007 (\href{http://www.comlab.ox.ac.uk/files/666/RR-07-05.pdf}{pdf})

\end{itemize}
This is further explored in

\begin{itemize}%
\item [[Bob Coecke]], Eric Paquette, [[Dusko Pavlovic]], \emph{Classical and quantum structures} (\href{http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf}{pdf})


\item [[Bob Coecke]], [[Chris Heunen]], \emph{Pictures of complete positivity in arbitrary dimension}, EPTCS 95, 2012, pp. 27-35 (\href{http://arxiv.org/abs/1110.3055}{arXiv:1110.3055})



\end{itemize}
\begin{itemize}%
\item [[Bob Coecke]], [[Chris Heunen]], [[Aleks Kissinger]], \emph{Categories of Quantum and Classical Channels} (\href{http://arxiv.org/abs/1305.3821}{arXiv:1305.3821})

\end{itemize}
For the universal property, see

\begin{itemize}%
\item Mathieu Huot, Sam Staton, \emph{Universal properties in quantum theory} (QPL 2018) (\href{https://www.mathstat.dal.ca/qpl2018/papers/QPL_2018_paper_68.pdf}{pdf}).

\end{itemize}
[[!redirects quantum channel]] [[!redirects quantum channels]] [[!redirects quantum operation]] [[!redirects quantum operations]] [[!redirects quantum operations and channels]]

[[!redirects completely positive map]] [[!redirects completely positive maps]]



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