update

Author: mhjensen

doc/pub/week5/html/week5-bs.html

@@ -615,7 +615,7 @@ <h2 id="binary-example" class="anchor">Binary example </h2>
 
 <p>If we now use the above binary example, we have (using that \( p_1=p \) and \( p_2=1-p \))</p>
 $$
-S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{1-p}.
+S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{(1-p)}.
 $$
 
 <p>For \( p=0 \) we have \( S=0 \) and for \( p=1 \) we get \( S=0 \). Thus, for the two
@@ -1004,14 +1004,14 @@ <h2 id="further-parts-of-proof" class="anchor">Further parts of proof </h2>
 we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
 decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
 of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
-(diagonal) singular values \( \sigma_0\leq \sigma_1 \leq \dots 0 \) as
+(diagonal) singular values \( 0\leq \sigma_0\leq \sigma_1 \leq \dots \sigma_{d-1} \) as
 </p>
 
 $$
 \boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
 $$
 
-
+<p>Note that the singular values can organized either as a descending series or as an ascending series.</p>
 <!-- !split -->
 <h2 id="svd-parts-in-proof" class="anchor">SVD parts in proof </h2>
 
@@ -1047,13 +1047,16 @@ <h2 id="different-dimensionalities" class="anchor">Different dimensionalities </
 <p>It is straight forward to prove this relation in case systems \( A \) and
 \( B \) have different dimensionalities.  Once we know the Schmidt
 decomposition of a state, we can immmediately say whether it is
-entangled or not. If a state \( \psi \) has is entangled, then its Schmidt
+entangled or not. If a state \( \psi \) is entangled, then its Schmidt
 decomposition has more than one term. Stated differently, the state is
-entangled if the so-called Schmidt rank is is greater than one.  There
+entangled if the so-called Schmidt rank is greater than one.
+</p>
+
+<p>There
 is another important property of the Schmidt decomposition which is
 related to the properties of the density matrices and their trace
-operations and the entropies. In order to introduce these concepts let
-us look at the two-qubit Hamiltonian described here.
+operations and the entropies. These concepts can be studied for example
+by studying the two-qubit Hamiltonian described above.
 </p>
 
 <!-- !split -->

doc/pub/week5/html/week5-reveal.html

@@ -635,7 +635,7 @@ <h2 id="binary-example">Binary example </h2>
 <p>If we now use the above binary example, we have (using that \( p_1=p \) and \( p_2=1-p \))</p>
 <p>&nbsp;<br>
 $$
-S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{1-p}.
+S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{(1-p)}.
 $$
 <p>&nbsp;<br>
 
@@ -1089,14 +1089,16 @@ <h2 id="further-parts-of-proof">Further parts of proof </h2>
 we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
 decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
 of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
-(diagonal) singular values \( \sigma_0\leq \sigma_1 \leq \dots 0 \) as
+(diagonal) singular values \( 0\leq \sigma_0\leq \sigma_1 \leq \dots \sigma_{d-1} \) as
 </p>
 
 <p>&nbsp;<br>
 $$
 \boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
 $$
 <p>&nbsp;<br>
+
+<p>Note that the singular values can organized either as a descending series or as an ascending series.</p>
 </section>
 
 <section>
@@ -1142,13 +1144,16 @@ <h2 id="different-dimensionalities">Different dimensionalities </h2>
 <p>It is straight forward to prove this relation in case systems \( A \) and
 \( B \) have different dimensionalities.  Once we know the Schmidt
 decomposition of a state, we can immmediately say whether it is
-entangled or not. If a state \( \psi \) has is entangled, then its Schmidt
+entangled or not. If a state \( \psi \) is entangled, then its Schmidt
 decomposition has more than one term. Stated differently, the state is
-entangled if the so-called Schmidt rank is is greater than one.  There
+entangled if the so-called Schmidt rank is greater than one.
+</p>
+
+<p>There
 is another important property of the Schmidt decomposition which is
 related to the properties of the density matrices and their trace
-operations and the entropies. In order to introduce these concepts let
-us look at the two-qubit Hamiltonian described here.
+operations and the entropies. These concepts can be studied for example
+by studying the two-qubit Hamiltonian described above.
 </p>
 </section>
 

doc/pub/week5/html/week5-solarized.html

@@ -565,7 +565,7 @@ <h2 id="binary-example">Binary example </h2>
 
 <p>If we now use the above binary example, we have (using that \( p_1=p \) and \( p_2=1-p \))</p>
 $$
-S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{1-p}.
+S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{(1-p)}.
 $$
 
 <p>For \( p=0 \) we have \( S=0 \) and for \( p=1 \) we get \( S=0 \). Thus, for the two
@@ -954,14 +954,14 @@ <h2 id="further-parts-of-proof">Further parts of proof </h2>
 we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
 decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
 of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
-(diagonal) singular values \( \sigma_0\leq \sigma_1 \leq \dots 0 \) as
+(diagonal) singular values \( 0\leq \sigma_0\leq \sigma_1 \leq \dots \sigma_{d-1} \) as
 </p>
 
 $$
 \boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
 $$
 
-
+<p>Note that the singular values can organized either as a descending series or as an ascending series.</p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="svd-parts-in-proof">SVD parts in proof </h2>
 
@@ -997,13 +997,16 @@ <h2 id="different-dimensionalities">Different dimensionalities </h2>
 <p>It is straight forward to prove this relation in case systems \( A \) and
 \( B \) have different dimensionalities.  Once we know the Schmidt
 decomposition of a state, we can immmediately say whether it is
-entangled or not. If a state \( \psi \) has is entangled, then its Schmidt
+entangled or not. If a state \( \psi \) is entangled, then its Schmidt
 decomposition has more than one term. Stated differently, the state is
-entangled if the so-called Schmidt rank is is greater than one.  There
+entangled if the so-called Schmidt rank is greater than one.
+</p>
+
+<p>There
 is another important property of the Schmidt decomposition which is
 related to the properties of the density matrices and their trace
-operations and the entropies. In order to introduce these concepts let
-us look at the two-qubit Hamiltonian described here.
+operations and the entropies. These concepts can be studied for example
+by studying the two-qubit Hamiltonian described above.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>

doc/pub/week5/html/week5.html

@@ -642,7 +642,7 @@ <h2 id="binary-example">Binary example </h2>
 
 <p>If we now use the above binary example, we have (using that \( p_1=p \) and \( p_2=1-p \))</p>
 $$
-S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{1-p}.
+S = -\sum_{i=1}^{2}p_i \log_2{p_i}=-p\log_2{p}-(1-p)\log_2{(1-p)}.
 $$
 
 <p>For \( p=0 \) we have \( S=0 \) and for \( p=1 \) we get \( S=0 \). Thus, for the two
@@ -1031,14 +1031,14 @@ <h2 id="further-parts-of-proof">Further parts of proof </h2>
 we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
 decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
 of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
-(diagonal) singular values \( \sigma_0\leq \sigma_1 \leq \dots 0 \) as
+(diagonal) singular values \( 0\leq \sigma_0\leq \sigma_1 \leq \dots \sigma_{d-1} \) as
 </p>
 
 $$
 \boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
 $$
 
-
+<p>Note that the singular values can organized either as a descending series or as an ascending series.</p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="svd-parts-in-proof">SVD parts in proof </h2>
 
@@ -1074,13 +1074,16 @@ <h2 id="different-dimensionalities">Different dimensionalities </h2>
 <p>It is straight forward to prove this relation in case systems \( A \) and
 \( B \) have different dimensionalities.  Once we know the Schmidt
 decomposition of a state, we can immmediately say whether it is
-entangled or not. If a state \( \psi \) has is entangled, then its Schmidt
+entangled or not. If a state \( \psi \) is entangled, then its Schmidt
 decomposition has more than one term. Stated differently, the state is
-entangled if the so-called Schmidt rank is is greater than one.  There
+entangled if the so-called Schmidt rank is greater than one.
+</p>
+
+<p>There
 is another important property of the Schmidt decomposition which is
 related to the properties of the density matrices and their trace
-operations and the entropies. In order to introduce these concepts let
-us look at the two-qubit Hamiltonian described here.
+operations and the entropies. These concepts can be studied for example
+by studying the two-qubit Hamiltonian described above.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>