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\title[CP-Invariance and Complete Positivity]{CP-Invariance and Complete Positivity \\ in Quantum Information Theory}
\author[Nathaniel Johnston]{Nathaniel Johnston \\ D. W. Kribs, V. I. Paulsen, R. Pereira, and E. St\o rmer}
\institute{University of Guelph}
\date{December 10, 2011}
\begin{document}
% Title Page and Outline
\frame{\titlepage}
\frame{
\frametitle{Order of Events}
\begin{itemize}
\item Operator Systems on Complex Matrices\medskip
\uncover<2->{
\item Positive and Completely Positive Maps\medskip
}
\uncover<3->{
\item Operator Systems $\leftrightarrow$ Right-CP-Invariant Cones\medskip
}
\uncover<4->{
\item Minimal and Maximal Operator Systems\medskip
}
\uncover<5->{
\item Other CP-Invariant Cones from Quantum Information Theory
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and Notation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cones and Completely Positive Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{What are Cones and Completely Positive Maps?}
We will be mostly concerned with the space of $n \times n$ complex matrices $M_n$.\medskip
\begin{itemize}
\uncover<2->{
\item A \emph{cone} $C \subset M_n$ is a set of Hermitian operators that is invariant under positive scaling.\medskip
}
\uncover<3->{
\item A cone $C$ is called \emph{convex} if $X + Y \in C$ whenever $X, Y \in C$.
}
\end{itemize}
}
\frame{
\frametitle{What are Cones and Completely Positive Maps?}
We can also consider cones of linear maps acting on $M_n$ (i.e., in $\mathcal{L}(M_n)$).\medskip
\begin{itemize}
\uncover<2->{
\item A \emph{cone} $C \subset \mathcal{L}(M_n)$ is a set of Hermiticity-preserving maps that is invariant under positive scaling.\medskip
}
\uncover<3->{
\item An \emph{adjoint map} is a map of the form $X \mapsto A^\dagger XA$, where $A^\dagger$ denotes the adjoint operator of $A$. We will denote this map by ${\rm Ad}_A$.\medskip
}
\uncover<4->{
\item The set of adjoint maps is a cone. Its convex hull is the cone of \emph{completely positive maps}.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Operator Systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{What is an Operator System?}
An (abstract) \emph{operator system on $M_n$} is a family of convex cones $C_m \subseteq M_m \otimes M_n$ (one cone for each $m \in \mathbb{N}$) that satisfy two properties:\medskip
\begin{enumerate}[1.]
\uncover<2->{
\item $C_1 = M_n^{+}$, the cone of positive-semidefinite operators in $M_n$; and\medskip
}
\uncover<3->{
\item for each $m_1,m_2 \in \bb{N}$ and $A \in M_{m_1,m_2}$ we have $({\rm Ad}_{A} \otimes id_n)(C_{m_1}) \subseteq C_{m_2}$.\medskip
}
\end{enumerate}
\uncover<4->{
Intuitively, these restrictions say that each cone $C_m$ is somehow ``like'' the cone $M_n^+$.
}
}
\frame{
\frametitle{Examples of Operator Systems on $M_n$}
Quantum information theorists are actually familiar with some operator systems on $M_n$ already.
\begin{itemize}
\uncover<2->{
\item The most natural operator system on $M_n$ is the one that arises by making the association $M_m \otimes M_n \cong M_{mn}$ in the natural way and letting $C_m = M_{mn}^+$; the cone of positive semidefinite operators. \textcolor{red}{Keep this ``na\"{i}ve'' operator system in mind!}\medskip}
\uncover<3->{
\item We will denote this ``na\"{i}ve'' operator system simply by $M_n$. We will denote other general operator systems on $M_n$ by things like $O_1$ and $O_2$.}
\end{itemize}
}
\frame{
\frametitle{Examples of Operator Systems on $M_n$}
\begin{itemize}
\item An operator $X \in M_m \otimes M_n$ is called \emph{separable} if it can be written as
\begin{align*}
X = \sum_i p_i\ketbra{\psi_i}{\psi_i} \otimes \ketbra{\phi_i}{\phi_i}, \quad p_i \geq 0 \ \forall \, i.
\end{align*}
\uncover<2->{\noindent \hspace*{-0.05in} If we let $S_m$ be the cone of separable operators in $M_m \otimes M_n$, then $\{S_m\}$ is an operator system (which we will denote by $OMAX$).\medskip}
\uncover<3->{
\item An operator $X \in M_m \otimes M_n$ is called \emph{block positive} if
\begin{align*}
(\bra{a} \otimes \bra{b})X(\ket{a} \otimes \ket{b}) \geq 0 \quad \text{ for all } \quad \ket{a} \in \mathbb{C}^m, \ \ket{b} \in \mathbb{C}^n.
\end{align*}}
\uncover<4->{\noindent \hspace*{-0.1in} If we let $P_m$ be the cone of block positive operators in $M_m \otimes M_n$, then $\{P_m\}$ is an operator system (which we will denote by $OMIN$).}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Positive and Completely Positive Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Positive Maps}
A map on $M_n$ is called \emph{positive} if it preserves positive semidefiniteness. That is, $\Phi \in \mathcal{L}(M_n)$ is positive if $\Phi(X) \in M_n^+$ whenever $X \in M_n^+$.\medskip
\begin{itemize}
\uncover<2->{
\item Positive maps play an important role in quantum information theory (particularly in entanglement theory).\medskip
}
\uncover<3->{
\item We will denote the set of postive maps by $\cl{P}$.\medskip
}
\uncover<4->{
\item We often want a stronger condition than positivity when dealing with operator systems though. We don't just want the cones $C_1$ to be preserved, but we want {\it all} of the cones $\{C_m\}$ to be preserved.
}
\end{itemize}
}
\frame{
\frametitle{Completely Positive Maps}
Suppose we are given operator systems $O_1$ and $O_2$, defined by cones $\{C_m\}$ and $\{D_m\}$, respectively.\medskip
\begin{itemize}
\uncover<2->{
\item A map $\Phi \in \mathcal{L}(M_n)$ is called \emph{completely positive from $O_1$ to $O_2$} if
\begin{align*}
(id_m \otimes \Phi)(C_m) \subseteq D_m \quad \text{ for all $m$}.
\end{align*}
}\vspace*{-0.2in}
\uncover<3->{
\item The set of completely positive maps from $O_1$ to $O_2$ will be denoted by $\mathcal{CP}(O_1,O_2)$.\medskip
}
\uncover<4->{
\item $\mathcal{CP}(M_n,M_n)$ is the usual set of completely positive maps that quantum information theorists know and love. For brevity, we will denote it simply by $\cl{CP}$.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operator Systems and Right-CP-Invariant Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Uniqueness of Operator Systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Uniqueness of Operator Systems}
The main workhorse that allows us to deal with operator systems is the following result, which says that they are uniquely determined by the $n$th cone -- that is, the cone $C_n \subset M_n \otimes M_n$.\smallskip
\uncover<2->{
\begin{prop}\label{prop:cn_to_os}
Let $C_n \subseteq M_n \otimes M_n$ be a convex cone such that
\begin{itemize}
\item $S_n \subseteq C_n \subseteq P_n$; and
\item $({\rm Ad}_A \otimes id_n)(C_n) \subseteq C_n$ for all $A \in M_n$.
\end{itemize}
Then there exists a unique family of cones $\{C_m\}_{m\neq n}$ such that $\{C_m\}_{m=1}^{\infty}$ defines an operator system on $M_n$.
\end{prop}}
}
\frame{
\frametitle{Uniqueness of Operator Systems}
\begin{cor}\label{cor:cn_to_os}
Let $\Phi : M_n \rightarrow M_n$ and let $O_1$ and $O_2$ be operator systems defined by families of cones $\{C_m\}_{m=1}^{\infty}$ and $\{D_m\}_{m=1}^{\infty}$, respectively. Then $\Phi \in \cl{CP}(O_1,O_2)$ if and only if $(id_n \otimes \Phi)(C_n) \subseteq D_n$.
\end{cor}\bigskip
\uncover<2->{
The above result generalizes the well-known result of Choi that says that $\Phi$ is completely positive if and only if it is $n$-positive.
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Right-CP-Invariant Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{What is a Right-CP-Invariant Cone?}
Observe that if $\Phi_1 \in \mathcal{CP}(O_1,O_2)$ and $\Phi_2 \in \mathcal{CP}(O_2,O_3)$, then $\Phi_2 \circ \Phi_1 \in \mathcal{CP}(O_1,O_3)$.\medskip
\begin{itemize}
\uncover<2->{
\item In particular, if $\Phi_1 \in \mathcal{CP}$ and $\Phi_2 \in \mathcal{CP}(M_n,O)$, then $\Phi_2 \circ \Phi_1 \in \mathcal{CP}(M_n,O)$.\medskip
}
\uncover<3->{
\item We use the term \emph{right-CP-invariant} for cones $\mathcal{C} \subseteq \cl{L}(M_n)$ with the property that $\mathcal{C} \circ \mathcal{CP} = \mathcal{C}$.
}
\end{itemize}
}
\frame{
\frametitle{Left-CP-Invariant and Mapping Cones}
We just saw that $\mathcal{CP}(M_n,O)$ is always right-CP-invariant.\medskip
\begin{itemize}
\uncover<2->{
\item Similarly, $\mathcal{CP}(O,M_n)$ is always \emph{left-CP-invariant}.\medskip
}
\uncover<3->{
\item A cone that is both left-CP-invariant and right-CP-invariant is called a \emph{mapping cone} -- such cones are well-studied for independent reasons in operator theory.\medskip
}
\uncover<4->{
\item Examples of mapping cones include the cones of positive, completely positive, and entanglement-breaking maps (denoted by $\cl{P}$, $\cl{CP}$, and $\cl{S}$, respectively).
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Operator Systems $\leftrightarrow$ Right-CP-Invariant Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Operator Systems $\leftrightarrow$ Right-CP-Invariant Cones}
Recall that $\mathcal{CP}(M_n,O)$ is always right-CP-invariant. It turns out that right-CP-invariance completely characterizes the possible cones of completely positive maps.\smallskip
\uncover<2->{
\begin{thm}\label{thm:right_cp_invariant}
Let $\cl{C} \subseteq \cl{L}(M_n)$ be a convex cone. The following are equivalent:
\begin{itemize}
\item $\cl{C}$ is right-CP-invariant with $\cl{S} \subseteq \cl{C} \subseteq \cl{P}$.
\item There exists an operator system $O$ such that $\cl{C} = \cl{CP}(M_n,O)$.
\end{itemize}
Furthermore, if $O$ is defined by the cones $\{C_m\}_{m=1}^{\infty}$ then $C_n$ is the cone of Choi matrices of maps in $\cl{C}$. Thus, $O$ is uniquely determined by $\cl{C}$.
\end{thm}}
}
\frame{
\frametitle{Basic Examples}
The previous theorem established a bijection between operator systems and right-CP-invariant cones, and furthermore showed that bijection is essentially the same as the Choi-Jamio\l kowski isomorphism.\medskip
\begin{itemize}
\uncover<2->{
\item The ``na\"{i}ve'' operator system $M_n$ corresponds to the cone of ``standard'' completely positive maps $\cl{CP}(M_n,M_n)$.\medskip
}
\uncover<3->{
\item The operator system $OMIN$, defined by the cones of block-positive operators $\{P_m\}$, corresponds to the cone of positive maps $\cl{P}$. $OMIN$ is the largest possible operator system.\medskip
}
\uncover<4->{
\item The operator system $OMAX$, defined by the cones of separable operators $\{S_m\}$, corresponds to the cone of entanglement-breaking maps $\cl{S}$. $OMAX$ is the smallest possible operator system.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Further Examples from Quantum Information Theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$k$-Positive and $k$-Entanglement Breaking Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{$k$-Positive Maps}
A linear map $\Phi \in \cl{L}(M_n)$ is called \emph{$k$-positive} if $id_k \otimes \Phi$ is positive.\medskip
\begin{itemize}
\uncover<2->{
\item $k$-positive maps play a role in entanglement theory analogous to the role of positive maps -- they help distinguish states with Schmidt number $\leq k$ from those with Schmidt number $> k$.\medskip
}
\uncover<3->{
\item The cone of $k$-positive maps is right-CP-invariant. The cones that define the corresponding operator system are the cones of \emph{$k$-block positive} operators. That is, operators such that $\bra{v}X\ket{v} \geq 0$ whenever the Schmidt rank of $\ket{v}$ is $\leq k$.\medskip
}
\uncover<4->{
\item This operator system is called the \emph{super $k$-minimal} operator system, denoted $OMIN_k$. It is the largest operator system $\{C_m\}$ such that $C_m = M_{mn}^+$ for $1 \leq m \leq k$.
}
\end{itemize}
}
\frame{
\frametitle{$k$-Entanglement Breaking Maps}
A linear map $\Phi \in \cl{L}(M_n)$ is called \emph{$k$-entanglement breaking} if $(id_n \otimes \Phi)(X)$ has Schmidt number $\leq k$ for any $X \in M_{n^2}^+$.\medskip
\begin{itemize}
\uncover<2->{
\item The cone of $k$-entanglement breaking maps is right-CP-invariant. The operator system associated with the cone of $k$-entanglement breaking maps is defined by the cones of operators with Schmidt rank $\leq k$.\medskip
}
\uncover<3->{
\item This operator system is called the \emph{super $k$-maximal} operator system, denoted $OMAX_k$. It is the smallest operator system $\{C_m\}$ such that $C_m = M_{mn}^+$ for $1 \leq m \leq k$.
}
\end{itemize}
} % draw op system tree at this point
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Anti-Degradable and Local $k$-Broadcasting Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Anti-Degradable Maps}
A map $\Phi \in \cl{CP}$ is called \emph{anti-degradable} if there exists $\Psi \in \cl{CP}$ such that $\Psi \circ \Phi^{C} = \Phi$, where $\Phi^{C}$ is the complementary map of $\Phi$.\medskip
\begin{itemize}
\uncover<2->{
\item The cone of anti-degradable maps is convex and right-CP-invariant.\medskip
}
\uncover<3->{
\item This cone is {\it not} left-CP-invariant.\medskip
}
\uncover<4->{
\item The operator system it gives rise to does not seem to be one that has been studied in operator theory.
}
\end{itemize}
}
\frame{
\frametitle{Anti-Degradable Maps}
An operator $X \in (M_m \otimes M_n)^+$ is called \emph{shareable} if there exists $\tilde{X} \in (M_m \otimes M_n \otimes M_n)^+$ such that $\Tr_2(\tilde{X}) = \Tr_3(\tilde{X}) = X$. We will denote by $H_m \subset M_m \otimes M_n$ the set of shareable operators.\medskip
\begin{itemize}
\uncover<2->{
\item The Choi matrix of an anti-degradable map is a shareable operator, and a shareable operator is always the Choi matrix of some anti-degradable map.\medskip
}
\uncover<3->{
\item The operator system associated with the cone of anti-degradable maps is $\{H_m\}$.
}
\end{itemize}
}
\frame{
\frametitle{Local $k$-Broadcasting Maps}
A map $\Phi \in \cl{CP}$ is called \emph{local $k$-broadcasting} if there exists a completely positive $\tilde{\Phi} : M_n \rightarrow M_n^{\otimes k}$ with the property that $\Phi = \Tr_{\overline{i}} \circ \tilde{\Phi}$ for all $1 \leq i \leq k$.\medskip
\begin{itemize}
\uncover<2->{
\item In the $k = 2$ case, these are exactly the anti-degradable maps (slightly non-trivial, but not horribly so).\medskip
}
\uncover<3->{
\item These cones, like the cone of anti-degradable maps, are convex and right-CP-invariant.\medskip
}
\uncover<4->{
\item These cones are {\it not} left-CP-invariant (except in the trivial $k = 1$ and $k = \infty$ cases, in which case we get the cones of completely positive and entanglement breaking maps, respectively).
}
\end{itemize}
}
\frame{
\frametitle{Local $k$-Broadcasting Maps}
An operator $X \in (M_m \otimes M_n)^+$ is called \emph{$k$-shareable} if there exists $\tilde{X} \in (M_m \otimes M_n^{\otimes k})^+$ such that $\Tr_{\overline{i}}(\tilde{X}) = X$ for all $2 \leq i \leq k+1$. We will denote by $H_m^{k} \subset M_m \otimes M_n$ the set of $k$-shareable operators.\medskip
\begin{itemize}
\uncover<2->{
\item The Choi matrix of a local $k$-broadcasting map is a $k$-shareable operator, and a $k$-shareable operator is always the Choi matrix of some local $k$-broadcasting map.\medskip
}
\uncover<3->{
\item The operator system associated with the cone of local $k$-broadcasting maps is $\{H_m^k\}$.\medskip
}
\uncover<4->{
\item These operator systems provide a natural (infinite) hierarchy that interpolates between $M_n$ and $OMAX$.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Open Questions and Further Reading}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Open Questions}
\frame{
\frametitle{Open Questions}
\begin{enumerate}[1.]
\item What other ``natural'' convex cones are right-CP-invariant?\smallskip
\begin{enumerate}[a)]\item A possible example is the cone of \emph{locally entanglement-annihilating} maps: maps $\Phi$ such that $(\Phi \otimes \Phi)(X) \in S_n$ for all $X \in (M_n \otimes M_n)^+$. This cone is right-CP-invariant (and left-CP-invariant), but is it convex?\medskip
\uncover<2->{
\item What about the set of entanglement binding maps? This set is right-CP-invariant. However, it is convex (and hence corresonds to an operator system) if and only if all bound entangled states are PPT (decade-old open problem).\medskip
}\end{enumerate}
\uncover<3->{
\item This seems to be a strong link between quantum information theory and operator theory. Find applications going in either direction!
}
\end{enumerate}
}
\subsection{Further Reading}
\frame{
\frametitle{Further Reading}
\begin{thebibliography}{99}
\bibitem{B1} N. J., D.~W. Kribs, V.~I. Paulsen, and R. Pereira, {\it Minimal and maximal operator spaces and operator systems in entanglement theory}. J. Funct. Anal. {\bf 260} 8, 2407-–2423 (2011).\\ E-print: arXiv:1010.1432 [math.OA]\bigskip
\bibitem{B2} N. J. and E. St\o rmer, {\it Mapping cones are operator systems}. Preprint (2011).\\ E-print: arXiv:1102.2012 [math.OA]
\end{thebibliography}
}
\end{document}