final version

matlab_notebook.tex

@@ -135,7 +135,7 @@ \section*{Generalized Bell states (Examples from \cite{Ghosh04})}
 In the code that follows, the function \texttt{GenPauli(a,b,n)} generates the 
 matrix corresponding to the operator $W_{a,b}\in\Unitary(\complex^{n})$ 
 defined in Eq.~\eqref{eq:generalized-Pauli-operators}.
-\subsection*{$5$ states in $\complex^{5}\otimes\complex^{5}$}
+\subsection*{$5$ maximally entangled states in $\complex^{5}\otimes\complex^{5}$}
 \begin{verbatim}
 n = 5;
 gen_bells = [vec(GenPauli(0,0,n)), ...
@@ -153,7 +153,7 @@ \subsection*{$5$ states in $\complex^{5}\otimes\complex^{5}$}
     0.9898
 \end{verbatim} \color{black}
 
-\subsection*{$6$ states in $\complex^{6}\otimes\complex^{6}$}
+\subsection*{$6$ maximally entangled states in $\complex^{6}\otimes\complex^{6}$}
 \begin{verbatim}
 n = 6;
 gen_bells = [vec(GenPauli(0,0,n)), ...

programs.tex

@@ -426,15 +426,13 @@ \subsubsection{Decomposable operator}
 A \emph{decomposable} map $\Phi:\Lin(\Y)\rightarrow\Lin(\X)$ is a linear map that can 
 be represented as the sum of a completely positive map and a completely 
 co-positive map, that is, there exist two completely positive maps 
-$\Psi,\Xi:\Lin(\Y)\rightarrow\Lin(\X)$, such that, for any $Y\in\Lin(\Y)$,
+$\Psi,\Xi:\Lin(\Y)\rightarrow\Lin(\X)$, such that
 \begin{equation}
-  \Phi(Y) = \Psi(Y) + (\pt {\circ}\, \Xi)(Y),
+  \Phi = \Psi + \pt {\circ}\, \Xi,
 \end{equation}
 where $\pt$ denotes the transpose map.
 \end{definition}
 
-
-
 \subsubsection{Exploiting symmetries}
 
 Whenever the ensemble of states we wish to distinguish exhibits some symmetry,
@@ -662,6 +660,22 @@ \subsubsection{Block-positive operators}
   J(\Phi) \in \BPos(\X:\Y).
 \end{equation}
 \end{itemize}
+In order to prove the equivalence, observe the following equation:
+\begin{equation}
+\label{eq:choi-positive}
+  (\I_{\X}\otimes y^{\ast})J(\Phi)(\I_{X}\otimes y) = \Phi(\overline{y}y^{\t}),
+\end{equation}
+which holds for every vector $y\in\Y$.
+Since the operator $\overline{y}y^{\t}$ is positive semidefinite,
+we have that (a) implies (b).
+To see the converse, recall that every positive semidefinite operator can be 
+written as a positive linear combination of rank-$1$ positive semidefinite 
+operators. Therefore, from the property (b), it holds that 
+\begin{equation}
+ (\I_{\X}\otimes y^{\t})J(\Phi)(\I_{X}\otimes \overline{y}) = \Phi(yy^{\ast}) 
+  \in\Pos(\X),
+\end{equation}
+for every vector $y\in\Y$, and thus by linearity $\Phi$ is a positive mapping. 
 
 Dual to the fact that there are non-separable positive semidefinite operators 
 is the fact that there are block-positive operators, which are not 
@@ -1107,7 +1121,7 @@ \section{Unambiguous state discrimination}
       & y_{i,j} \in \real, \quad 1 \leq i,j \leq N, \quad i \neq j.\notag
     \end{align}
 \end{center}
-In the next chapter we will see an application of this program for unambiguous 
-PPT-distinguishability. In particular, we will analyze a set of states where 
+In the next chapter we will see an application of this program to the problem of 
+unambiguous PPT-distinguishability. In particular, we will analyze a set of states where 
 the unambiguous PPT-distinsuishability is strictly lower than the regular 
 PPT-distinguishability calculated by the program of Section~\ref{sec:ppt-measurements-program}. 

thesis.brf

@@ -67,14 +67,14 @@
 \backcite {Doherty02,Doherty04}{{39}{4.2.1}{subsection.4.2.1}}
 \backcite {Piani06}{{42}{4.2.1}{section*.13}}
 \backcite {Gharibian13}{{45}{4.2.2}{subsection.4.2.2}}
-\backcite {Chruscinski2014}{{46}{4.2.2}{equation.4.2.44}}
-\backcite {Doherty02,Doherty04}{{47}{4.2.3}{subsection.4.2.3}}
-\backcite {Doherty02}{{48}{4.2.3}{equation.4.2.50}}
-\backcite {Myhr09}{{49}{4.2.3}{equation.4.2.52}}
+\backcite {Chruscinski2014}{{47}{4.2.2}{equation.4.2.46}}
+\backcite {Doherty02,Doherty04}{{48}{4.2.3}{subsection.4.2.3}}
+\backcite {Doherty02}{{49}{4.2.3}{equation.4.2.52}}
+\backcite {Myhr09}{{49}{4.2.3}{equation.4.2.54}}
 \backcite {Gharibi10}{{54}{4.4}{section.4.4}}
 \backcite {Doherty04}{{54}{4.4}{section.4.4}}
 \backcite {MartinGroetschel93}{{54}{4.4}{section.4.4}}
-\backcite {Johnston2015}{{55}{4.4}{equation.4.4.69}}
+\backcite {Johnston2015}{{55}{4.4}{equation.4.4.71}}
 \backcite {Eldar03}{{55}{4.5}{section.4.5}}
 \backcite {Yu12}{{57}{5}{chapter.5}}
 \backcite {Yu14}{{57}{5}{chapter.5}}