Minor edits

Advanced/Introduction to Quantum/src/parts/03 two qubits.tex

@@ -2,7 +2,7 @@
 
 
 \definition{}
-Just as before, we'll represent multi-quibit states as linear combinations of multi-qubit basis states. \par
+Just as before, we'll represent multi-qubit states as linear combinations of multi-qubit basis states. \par
 For example, a two-qubit state $\ket{ab}$ is the four-dimensional unit vector
 \begin{equation}
 	\begin{bmatrix}
@@ -33,7 +33,7 @@ we get one of the four basis states with the following probabilities:
 	\item $\mathcal{P}(\ket{10}) = c^2$
 	\item $\mathcal{P}(\ket{11}) = d^2$
 \end{itemize}
-Of course, the sum of all the above probabilities is $1$.
+As before, the sum of all the above probabilities is $1$.
 
 
 \problem{}

Advanced/Introduction to Quantum/src/parts/04 logic gates.tex

@@ -26,11 +26,11 @@ map, we can write it as follows:
 
 \definition{}
 Before we discussing multi-qubit quantum gates, we need to review to classical logic. \par
-Of course, a classical logic gate is a linear map from $\mathbb{B}^m$ to $\mathbb{B}^n$
+Of course, a classical logic gate is a linear map from $\{0,1\}^m$ to $\{0,1\}^n$
 
 
 \problem{}<notgatex>
-The \texttt{not} gate is a map from $\mathbb{B}$ to $\mathbb{B}$ defined by the following table: \par
+The \texttt{not} gate is a map defined by the following table: \par
 
 \begin{itemize}
 	\item $X\ket{0} = \ket{1}$

Advanced/Introduction to Quantum/src/parts/05 quantum gates.tex

@@ -10,7 +10,7 @@ satisfies $GG^\text{T} = I$. \par
 This implies the following: \par
 
 \begin{itemize}
-	\item $G$ is square \par
+	\item $G$ is square. In other words, it has as many rows as it has columns. \par
 	\note{
 		If we think of $G$ as a map, this means that $G$ has as many inputs as it has outputs. \\
 		This is to be expected: we stated earlier that quantum gates do not destroy or create qubits.
@@ -29,7 +29,7 @@ We can restate the above definition as follows: \par
 A quantum gate is an invertible map from $\mathbb{U}^n$ to $\mathbb{U}^n$.
 
 
-\definition{}<qgateislinear>
+\generic{Remark:}
 Let $G$ be a quantum gate. \par
 Since quantum gates are, by definition, \textit{linear} maps,
 the following holds: \par

Advanced/Introduction to Quantum/src/parts/07 superdense.tex

@@ -53,6 +53,7 @@ The $Z$ gate is defined as follows: \par
 \problem{}
 Suppose that Alice and Bob are each in possession of one qubit. \par
 These two qubits are entangled, and have the compound state $\ket{\Phi^+}$. \par
+\note[Note]{We could say that they each have \say{half} of $\ket{\Phi^+}$.}
 How can Alice send a two-bit classical state
 (i.e, one of the four values \texttt{00}, \texttt{01}, \texttt{10}, \texttt{11}) \par
 to Bob by only sending one qubit?