Update week7.do.txt

Author: mhjensen

doc/src/week7/week7.do.txt

@@ -623,7 +623,7 @@ remaining half to the eigenspace with eigenvalue $-1$.
 
 This term gives the correct eigenvalue when operating on the first
 qubit. In principle thus we don't need to rewrite string of operators.
-However, let us rewrite it via a unitary transformation in
+However, as discussed below as well, let us rewrite it via a unitary transformation in
 order to have $\bm{P}=\bm{Z}\otimes\bm{I}$.  To do so, consider the
 transformation
 
@@ -671,13 +671,18 @@ and $\vert 11\rangle$.
 ===== Transformations =====
 
 Any unitary transformation of such matrices also describes two
-half-spaces labeled with eigenvalues. For example, from the identity
-that . Similar to the one-qubit case, all two-qubit Pauli-measurements
-can be written as for unitary matrices . The transformations are
-enumerated in the following table.
-
-
+half-spaces labeled with eigenvalues. For example
+!bt
+\[
+\bm{X}\otimes\bm{X}=\bm{H}\otimes\bm{H}(\bm{Z}\otimes\bm{Z})\bm{H}\otimes\bm{H},, 
+\]
+!et
 
+follows from from the identity that $\bm{Z}=\bm{HXH}$.  Similar to the
+one-qubit case, all two-qubit Pauli-measurements can be written in
+terms of unitary transformations
+$\bm{U}^{\dagger}(\bm{Z}\otimes\bm{I})\bm{U}$ with $\bm{U}$ being
+$4\times 4$ unitary matrices.
 
 !split
 ===== More terms =====
@@ -695,18 +700,19 @@ which results in
 \[
 	\begin{bmatrix}
 		1 & 0 & 0 & 0 \\
-		0 & 0 & 0 & 1 \\
-		0 & 0 & 1 & 0 \\
-		0 & 1 & 0 & 0
-	\end{bmatrix}.
+		0 & 1 & 0 & 0 \\
+		0 & 0 & -1 & 0 \\
+		0 & 0 & 0 & -1
+	\end{bmatrix},
 \]
 !et
-
+which we recognize as our $\bm{Z}\otimes\bm{I}$ tensor product. 
 
 !split
 ===== Explicit expressions =====
-In order to perform our measurements for our simple two-qubit Hamiltonian
-we  need the following unitary transformations $\bm{U}$
+
+In order to perform our measurements for our simple two-qubit
+Hamiltonian we need the following unitary transformations $\bm{U}$
 !bt
 \begin{align*}
 \bm{Z}\otimes\bm{I}\hspace{1cm} & \bm{U}=\bm{I}\otimes\bm{I}\\
@@ -716,6 +722,7 @@ we  need the following unitary transformations $\bm{U}$
 \end{align*}
 !et
 
+These are the gates that are relevant for the simpler two-qubit Hamiltonian of project 1.
 !split
 ===== More complete list and derivations of expressions for strings of operators =====
 
@@ -729,6 +736,28 @@ For a two qubit system we list here the possible transformations
 \end{align*}
 !et
 
+
+!split
+===== Additional remarks =====
+
+Here, the CNOT operation appears for the following reason. Each Pauli measurement that doesn't include the 
+ matrix is equivalent up to a unitary to 
+ by the earlier reasoning. The eigenvalues of 
+ only depend on the parity of the qubits that comprise each computational basis vector, and the controlled-not operations serve to compute this parity and store it in the first bit. Then once the first bit is measured, you can recover the identity of the resultant half-space, which is equivalent to measuring the Pauli operator.
+
+Also, while it can be tempting to assume that measuring 
+ is the same as sequentially measuring 𝟙
+ and then 𝟙
+, this assumption would be false. The reason is that measuring 
+ projects the quantum state into either the 
+ or 
+ eigenstate of these operators. Measuring 𝟙
+ and then 𝟙
+ projects the quantum state vector first onto a half space of 𝟙
+ and then onto a half space of 𝟙
+. As there are four computational basis vectors, performing both measurements reduces the state to a quarter-space and hence reduces it to a single computational basis vector.
+
+
 !split
 ===== Lipkin model =====