Update week7.do.txt

Author: mhjensen

doc/src/week7/week7.do.txt

@@ -548,6 +548,11 @@ computational basis states, that is the states $\vert 00\rangle =\vert 0\rangle
 \times \vert 0\rangle$, $\vert 01 \rangle$, $\vert 10\rangle$ and
 $\vert 11\rangle$.
 
+This process is referred to in the language of Pauli measurements as
+_measuring Pauli-Z_" and is entirely equivalent to performing a
+computational basis measurement.
+
+
 !split
 ===== The specific eigenvalues =====
 
@@ -573,7 +578,7 @@ and
 	\end{bmatrix}\begin{bmatrix} 0\\ 0 \\ 1 \\ 0\end{bmatrix}=-1\begin{bmatrix} 0\\ 0 \\ 1 \\ 0\end{bmatrix}.
 \]
 !et
-We don't get the correct eigenvalues if we perform the measurement on the second qubits!
+We don't get the correct eigenvalues if we perform the measurement on the second qubit!
 
 !split
 ===== The $\bm{I}\otimes \bm{Z}$ term =====
@@ -609,6 +614,13 @@ correct eigenvalues when measured on the first qubit. Try this as an exercise.
 !split
 ===== The $\bm{Z}\otimes \bm{Z}$ term =====
 
+Thus the tensor products of two Pauli-$\bm{Z}$ operators forms a matrix
+composed of two spaces consisting of $+1$ and $-1$ as eigenvalues.
+As with the single-qubit case, both constitute a half-space, meaning that half of
+the accessible vector space belongs to the eigenspace with eigenvalue $+1$ and the
+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
@@ -655,6 +667,18 @@ To see this, act with $\bm{P}$ on the states $\vert 00\rangle =\vert
 0\rangle \times \vert 0\rangle$, $\vert 01 \rangle$, $\vert 10\rangle$
 and $\vert 11\rangle$.
 
+!split
+===== 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.
+
+
+
+
 !split
 ===== More terms =====