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\title{Right CP-Invariant Cones of Superoperators}
\author[Nathaniel Johnston]{Nathaniel Johnston \\\medskip based on joint work with\\ D.~W. Kribs, V.~I. Paulsen, R. Pereira, and E. St\o rmer}
\institute{MAO 2012, Harbin, China}
\date{July 14, 2012}
\begin{document}
% Title Page and Outline
\frame{\titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and Definitions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Choi Matrices}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Choi Matrices}
We will be mostly concerned with the space of $n \times n$ complex matrices $M_n$ and the space of linear maps on $M_n$: $\cl{L}(M_n)$.\bigskip
\uncover<2->{
Given a linear map $\Phi \in \cl{L}(M_n)$, its \emph{Choi matrix} is defined by
\begin{align*}
C_\Phi := \sum_{ij=1}^n E_{ij} \otimes \Phi(E_{ij}),
\end{align*}
where $\{E_{ij}\}$ is the family of standard matrix units in $M_n$.\bigskip
}
\uncover<3->{
Given a set $\cl{K} \subseteq \cl{L}(M_n)$, we define
\begin{align*}
C_{\cl{K}} := \big\{ C_{\Phi} : \Phi \in \cl{K} \big\}.
\end{align*}
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{What are Cones?}
A \emph{cone} $K \subset M_n$ is a set of Hermitian operators such that $\lambda K = K$ for all $\lambda \geq 0$.\bigskip
\uncover<2->{
Similarly, a set $\cl{K} \subset \mathcal{L}(M_n)$ is a \emph{cone} if $C_\cl{K}$ is a cone.\bigskip
}
\uncover<3->{
Equivalently, $\cl{K} \subset \mathcal{L}(M_n)$ is a cone if it is a set of Hermiticity-preserving maps (i.e., maps $\Phi$ such that $\Phi(X)^\dagger = \Phi(X)$ whenever $X^\dagger = X$) with the property that $\lambda \cl{K} = \cl{K}$ for all $\lambda \geq 0$.
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Positive and Completely Positive Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Positive Maps}
A map $\Phi$ on $M_n$ is called \emph{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 denote the set of positive maps by $\cl{P}$.\medskip
}
\uncover<4->{
\item $\cl{P}$ is a convex cone.
}
\end{itemize}
}
\frame{
\frametitle{Completely Positive Maps}
If $id_m \otimes \Phi$ is positive for all $m \geq 1$, then $\Phi$ is called \emph{completely positive} (CP).\medskip
\begin{itemize}
\uncover<2->{
\item A well-known result of Choi (1975) says that $\Phi$ is CP $\Leftrightarrow$ $C_\Phi \in (M_n \otimes M_n)^+$ $\Leftrightarrow$ there exist operators $\{A_i\}$ such that
\begin{align*}
\Phi(X) = \sum_i A_i X A_i^*.
\end{align*}\vspace*{-0.1in}
}
\uncover<3->{
\item We denote the set of completely positive maps by $\cl{CP}$.\medskip
}
\uncover<4->{
\item $\cl{CP}$ is a convex cone.
}
\end{itemize}
}
\frame{
\frametitle{Superpositive Maps}
If $\Phi$ can be written in the form\medskip
\begin{align*}
\Phi(X) = \sum_i A_i X A_i^*
\end{align*}
with ${\rm rank}(A_i) = 1$ for all $i$, then $\Phi$ is called \emph{superpositive}.\medskip
\begin{itemize}
\uncover<2->{
\item We denote the set of superpositive maps by $\cl{S}$.\medskip
}
\uncover<3->{
\item $\cl{S}$ is a convex cone.
}
\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 $K_m \subseteq M_m \otimes M_n$ (one cone for each $m \geq 1$) that satisfy two properties:\medskip
\begin{enumerate}[1.]
\uncover<2->{
\item $K_1 = M_n^{+}$, the cone of positive-semidefinite operators in $M_n$; and\medskip
}
\uncover<3->{
\item for each $m_1,m_2 \geq 1$ and $A \in M_{m_1,m_2}$ we have $(A \otimes I)K_{m_1}(A^* \otimes I) \subseteq K_{m_2}$.\medskip
}
\end{enumerate}
\uncover<4->{
Intuitively, these restrictions say that each cone $K_m$ is somehow ``like'' the cone of positive semidefinite operators.
}
}
\frame{
\frametitle{Examples of Operator Systems on $M_n$}
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 $K_m = M_{mn}^+$; the cone of positive semidefinite operators. \textcolor{red}{Keep this ``na\"{i}ve'' operator system in mind!}\medskip
\uncover<2->{
We denote the ``na\"{i}ve'' operator system simply by $M_n$. We will denote other general operator systems on $M_n$ by things like $O_1(M_n)$ and $O_2(M_n)$ (or simply $O_1$ and $O_2$).}
}
\frame{
\frametitle{Examples of Operator Systems on $M_n$}
An operator $X \in M_m \otimes M_n$ is called \emph{separable} if it can be written in the form
\begin{align*}
X = \sum_i A_i \otimes B_i \ \ \text{ with } \ \ A_i \in M_m^+, B_i \in M_n^+ \ \forall \, i.
\end{align*}
\uncover<2->{
If we let $S_m$ be the cone of separable operators in $M_m \otimes M_n$, then $\{S_m\}$ is an operator system.\bigskip
}
\uncover<3->{
Also, $S_n = C_{\cl{S}}$.
}
}
\frame{
\frametitle{Examples of Operator Systems on $M_n$}
An operator $X \in M_m \otimes M_n$ is called \emph{block positive} if
\begin{align*}
(\mathbf{a} \otimes \mathbf{b})^* X(\mathbf{a} \otimes \mathbf{b}) \geq 0 \quad \text{ for all } \quad \mathbf{a} \in \mathbb{C}^m, \ \mathbf{b} \in \mathbb{C}^n.
\end{align*}
\uncover<2->{
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.\bigskip
}
\uncover<3->{
Also, $P_n = C_{\cl{P}}$.
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Right CP-Invariant Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{What is a Right CP-Invariant Cone?}
We use the term \emph{right CP-invariant} for cones $\mathcal{K}$ with the property that $\mathcal{K} \circ \mathcal{CP} \subseteq \mathcal{K}$.\medskip
\begin{itemize}
\uncover<2->{
\item We can analogously define a \emph{left CP-invariant} cone $\cl{K}$ to be one with the property that $\mathcal{CP} \circ \mathcal{K} \subseteq \mathcal{K}$.\medskip
}
\uncover<3->{
\item The cones $\cl{P}$, $\cl{CP}$, and $\cl{S}$ are right (and left) CP-invariant.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Properties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Dual Cones}
We define the \emph{dual} of a cone $\cl{K} \subseteq \cl{L}(M_n)$ via Choi matrices:
\begin{align*}
\cl{K}^\circ := \big\{ \Phi \in \cl{L}(M_n) : {\rm Tr}(C_\Phi C_\Psi) \geq 0 \ \ \forall \, \Psi \in \cl{K} \big\}.
\end{align*}
\begin{itemize}
\uncover<2->{
\item $\cl{CP}^\circ = \cl{CP}$.\bigskip
}
\uncover<3->{
\item $\cl{P}^\circ = \cl{S}$.
}
\end{itemize}
}
\frame{
\frametitle{Properties of Right CP-Invariant Cones}
\begin{prop}
Let $\cl{K} \subseteq \cl{L}(M_n)$ be a right CP-invariant cone. The following are equivalent:
\begin{enumerate}[(a)]
\item $\Phi \in \cl{K}$; and
\item $(id_n \otimes \Phi)(X) \in C_{\cl{K}}$ for all $X \in (M_n \otimes M_n)^+$.
\end{enumerate}
\end{prop}\medskip
\begin{itemize}
\uncover<2->{
\item Special case: if $\cl{K} = \cl{S}$ then this says that $\Phi \in \cl{S}$ if and only if $(id_n \otimes \Phi)(X)$ is separable for all $X \in (M_n \otimes M_n)^+$.\medskip
}
\uncover<3->{
\item For this reason, superpositive maps are sometimes called \emph{entanglement-breaking maps}.
}
\end{itemize}
}
\frame{
\frametitle{Properties of Right CP-Invariant Cones}
\begin{prop}
If $\cl{K} \subseteq \cl{L}(M_n)$ is a right CP-invariant cone then so is $\cl{K}^\circ$.
\end{prop}
\medskip
\uncover<2->{
\begin{prop}
Let $\cl{K} \subseteq \cl{L}(M_n)$ be a right CP-invariant cone and let $\Phi \in \cl{L}(M_n)$. The following are equivalent:
\begin{enumerate}[(a)]
\item $\Phi \in \cl{K}^\circ$; and
\item $\Omega^\dagger \circ \Phi$ is completely positive for all $\Omega \in \cl{K}$.
\end{enumerate}
\end{prop}
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Completely Positive Maps on Operator Systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Completely Positive Maps}
Suppose we are given operator systems $O_1$ and $O_2$, defined by cones $\{J_m\}$ and $\{K_m\}$, respectively.\medskip
\begin{itemize}
\uncover<2->{
\item A map $\Phi$ is called \emph{completely positive from $O_1$ to $O_2$} if
\begin{align*}
(id_m \otimes \Phi)(J_m) \subseteq K_m \quad \text{ for all $m$}.
\end{align*}
}\vspace*{-0.2in}
\uncover<3->{
\item We denote the set of completely positive maps from $O_1$ to $O_2$ by $\mathcal{CP}(O_1,O_2)$.\medskip
}
\uncover<4->{
\item $\mathcal{CP}(M_n,M_n)$ is the usual set of ``standard'' completely positive maps.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Operator Systems $\leftrightarrow$ Right CP-Invariant Cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Operator Systems $\leftrightarrow$ Right CP-Invariant Cones}
It is easy to see that $\mathcal{CP}(M_n,O)$ is right CP-invariant for any operator system $O$. In fact, right CP-invariance completely characterizes the possible cones of completely positive maps.\smallskip
\uncover<2->{
\begin{thm}\label{thm:right_cp_invariant}
Let $\cl{K} \subseteq \cl{L}(M_n)$ be a convex cone. The following are equivalent:
\begin{itemize}
\item $\cl{K}$ is right CP-invariant with $\cl{S} \subseteq \cl{K} \subseteq \cl{P}$.
\item There exists an operator system $O$ such that $\cl{K} = \cl{CP}(M_n,O)$.
\end{itemize}
\end{thm}}
}
\frame{
\frametitle{Basic Examples}
We noted earlier that $\cl{P}$, $\cl{CP}$, and $\cl{S}$ are right CP-invariant. So what are the corresponding operator systems?\medskip
\begin{itemize}
\uncover<2->{
\item The ``na\"{i}ve'' operator system $M_n$ gives the cone of ``standard'' completely positive maps $\cl{CP}$ (by definition).\medskip
}
\uncover<3->{
\item The operator system $\{P_m\}$ of block positive operators gives the cone of positive maps $\cl{P}$.\medskip
}
\uncover<4->{
\item Paulsen, Todorov, and Tomforde (2010) showed that there is a largest operator system $OMIN(M_n)$. It is defined by the cones $\{P_m\}$.
}
\end{itemize}
}
\frame{
\frametitle{Basic Examples}
Similarly, the operator system $\{S_m\}$ of separable operators gives the cone of superpositive maps $\cl{S}$.\bigskip
\begin{itemize}
\uncover<2->{
\item Paulsen, Todorov, and Tomforde (2010) also showed that there is a smallest operator system $OMAX(M_n)$. It is defined by the cones $\{S_m\}$.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{More Examples from Quantum Information Theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$k$-Positive and $k$-Superpositive 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. The set of $k$-positive maps is $\cl{P}_k$.\bigskip
\uncover<2->{
A map is called \emph{$k$-superpositive} if it can be written in the form
\begin{align*}
\Phi(X) = \sum_i A_i X A_i^*
\end{align*}
with ${\rm rank}(A_i) \leq k$ for all $i$. The set of $k$-superpositive maps is $\cl{S}_k$.\bigskip
}
\uncover<3->{
$\cl{P}_k$ and $\cl{S}_k$ are right (and left) CP-invariant.
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Anti-Degradable Maps}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Complementary Maps}
Recall the Stinespring dilation theorem, which says that for any CP map $\Phi \in \cl{L}(M_n)$ there exists $A : \bb{C}^n \rightarrow \bb{C}^{n^2} \otimes \bb{C}^n$ so that:
\begin{align*}
\Phi(X) = \Tr_1(AXA^*).
\end{align*}
\uncover<2->{
The \emph{complementary map} of $\Phi$ is defined by
\begin{align*}
\Phi^{C}(X) := \Tr_2(AXA^*).
\end{align*}
}
}
\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 $\Psi \circ \Phi^{C} = \Phi$.\medskip
\begin{itemize}
\uncover<2->{
\item Intuitively, these maps leak more information than they preserve.\medskip
}
\uncover<3->{
\item The cone of anti-degradable maps is convex and right CP-invariant.\medskip
}
\uncover<4->{
\item This cone is {\it not} left CP-invariant.
}
\end{itemize}
}
\frame{
\frametitle{Shareable Operators}
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 $\Tr_2(\tilde{X}) = \Tr_3(\tilde{X}) = X$.\medskip
\begin{itemize}
\uncover<2->{
\item We use $H_m$ to denote the cone of shareable operators in $M_m \otimes M_n$.\medskip
}
\uncover<3->{
\item The set of cones $\{H_m\}$ forms an operator system, which we denote $O_H$.\medskip
}
\uncover<4->{
\item $\cl{CP}(M_n,O_H)$ is the set of anti-degradable maps.
}
\end{itemize}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Thank you!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thank you!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{
\frametitle{Thank you!}
\begin{center}\Large{Thank you!}\end{center}
}
\end{document}