Various fixes.

src/op/system/complete-upgrade.tex

@@ -31,7 +31,7 @@ The following is a slight generalisation of one form of the above example, but s
 
 \begin{proposition}
 \label{proposition:multiply-completely-bounded}
-    Let $A$ be a $C^*$-algebra, $S, T \subset A$ be operator systems, and $\phi: S \to T$. If there exists $x \in A$ such that $\phi(s) = x\phi(s)x^*$ for all $s \in S$, then $\phi$ is completely positive. 
+    Let $A$ be a $C^*$-algebra, $S, T \subset A$ be operator systems, and $\phi: S \to T$. If there exists $x \in A$ such that $\phi(s) = xsx^*$ for all $s \in S$, then $\phi$ is completely positive. 
 \end{proposition}
 
 \begin{proposition}
@@ -144,7 +144,7 @@ The following is a slight generalisation of one form of the above example, but s
     &= \sum_{i, j, k, l = 1}^n (E_{ij} \otimes a_{i, j})^*(E_{kl} \otimes a_{k, l}) = \sum_{i, j, k, l = 1}^n (E_{ji}E_{kl}) \otimes (a_{i, j}^* a_{k, l})
     \end{align*}
     
-    For each $1 \le i, j, k, l \le n$, $E_{ji}E_{kl} = E_{il}$ if $i = k$, and $E_{ji}E_{kl} = 0$ otherwise. Thus after relabeling, 
+    For each $1 \le i, j, k, l \le n$, $E_{ji}E_{kl} = E_{jl}$ if $i = k$, and $E_{ji}E_{kl} = 0$ otherwise. Thus after relabeling, 
     \[
         P = \sum_{j, k, l = 1}^nE_{jl} \otimes (a_{k, j}^*a_{k, l}) =  \sum_{k = 1}^n \sum_{i, j = 1}^n E_{ij} \otimes (a_{k, i}^*a_{k, j})
     \]
@@ -193,7 +193,7 @@ The following is a slight generalisation of one form of the above example, but s
     \sum_{k = 1}^n E_{kk} \otimes \phi(T) = \sum_{k = 1}^n S_k(T \otimes 1_B)S_k^*
     \]
 
-    What the above shows, is that, under the identification that
+    What the above shows is that, under the identification that
     \[
     M_n(\complex) \iso M_n(\complex) \otimes 1_B \subset M_n(B)
     \]
@@ -233,7 +233,7 @@ The following is a slight generalisation of one form of the above example, but s
 
     (Not Completely Positive): For each $1 \le i, j \le n$, $\text{Tr}(E_{ij}) = 1$ if and only if $i = j$. So the Choi matrix is given by
     \begin{align*}
-    C_{ij} &= \phi_n\paren{\sum_{i, j = 1}^n E_{ij} \otimes E_{ij}} = \sum_{i, j = 1}^n E_{ij} \otimes \phi(E_{ij}) \\
+    C_{\phi} &= \phi_n\paren{\sum_{i, j = 1}^n E_{ij} \otimes E_{ij}} = \sum_{i, j = 1}^n E_{ij} \otimes \phi(E_{ij}) \\
     &= \sum_{i, j = 1}^n E_{ij} \otimes (\one_{\bracsn{i = j}} \otimes 1_{M_n(\complex)} - E_{ij}) \\
     &= \sum_{i = 1}^n E_{ii} \otimes 1_{M_n(\complex)} - \sum_{i, j = 1}^n E_{ij} \otimes E_{ij} \\
     &= 1_{M_n(\complex)} \otimes 1_{M_n(\complex)}- \sum_{i, j = 1}^n E_{ij} \otimes E_{ij}
@@ -246,20 +246,20 @@ The following is a slight generalisation of one form of the above example, but s
 \label{proposition:choi-kraus}
     Let $\phi: M_n(\complex) \to M_m(\complex)$ be a completely positive map, then there exists $\bracsn{V_j}_1^{mn} \subset L(\complex^m; \complex^n)$ such that
     \[
-    \phi(a) = \sum_{j = 1}^{nk}V_j^* a V_j
+    \phi(a) = \sum_{j = 1}^{mn}V_j^* a V_j
     \]
 
     for all $a \in M_n(\complex)$. 
 \end{proposition}
 \begin{proof}
-    Let $(\pi, V, K)$ be a minimal Stinespring representation for $\phi$. Since $\phi(M_n(\complex))V\complex^m$ is dense in $K$, $K = \phi(M_n(\complex))V\complex^m$, and $\dim(K) \le n^2m$. 
+    Let $(\pi, V, K)$ be a minimal Stinespring representation for $\phi$. Since $\pi(M_n(\complex))V\complex^m$ is dense in $K$, $K = \pi(M_n(\complex))V\complex^m$, and $\dim(K) \le n^2m$. 
 
     As $M_n(\complex)$ is simple, the only (non-zero) irreducible representation of $M_n(\complex)$ is the identity. Since $(K, \pi)$ is a direct sum of irreducible representations, after appending additional factors, assume without loss of generality that $K = \bigoplus_{k = 1}^{nm}\complex^n$ and that $\pi(a) = I_{M_{mn}(\complex)} \otimes a$ for all $a \in M_n(\complex)$.
 
     For each $1 \le k \le mn$, let $P_k \in B(K)$ be the orthogonal projection onto the subspace corresponding to the $k$-th summand, and $Q_k \in L(K; \complex^n)$ be the orthogonal projection onto the $k$-th summand itself, then
     \begin{align*}
-    \phi(a) &= V^*\pi(a)V = \sum_{k = 1}^{mn}V^* (P_j\pi(a)P_j)V = \sum_{k = 1}^{mn}V^*Q_j^*aQ_jV \\
-    &= \sum_{k = 1}^{mn} (Q_jV)^* a (Q_jV)
+    \phi(a) &= V^*\pi(a)V = \sum_{k = 1}^{mn}V^* (P_k\pi(a)P_k)V = \sum_{k = 1}^{mn}V^*Q_k^*aQ_kV \\
+    &= \sum_{k = 1}^{mn} (Q_kV)^* a (Q_kV)
     \end{align*}
     
 \end{proof}