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‎doc/pub/week2/html/week2-bs.html‎

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@@ -117,32 +117,20 @@
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('Examples of entanglement', 2, None, 'examples-of-entanglement'),
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119119
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120-
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121-
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122-
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122+
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('Ex5: Reduced density operators I',
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('4: Reduced density operators II',
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('The next lecture, February 4, 2026',
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@@ -229,14 +217,11 @@
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<!-- navigation toc: --> <li><a href="#examples-of-entanglement" style="font-size: 80%;">Examples of entanglement</a></li>
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<!-- navigation toc: --> <li><a href="#ground-state-of-helium" style="font-size: 80%;">Ground state of helium</a></li>
231219
<!-- navigation toc: --> <li><a href="#maximally-entangled" style="font-size: 80%;">Maximally entangled</a></li>
232-
<!-- navigation toc: --> <li><a href="#density-matrices-in-more-detail" style="font-size: 80%;">Density matrices in more detail</a></li>
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<!-- navigation toc: --> <li><a href="#second-exercise-set" style="font-size: 80%;">Second exercise set</a></li>
234-
<!-- navigation toc: --> <li><a href="#ex1-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">Ex1: One-qubit basis and Pauli matrices</a></li>
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<!-- navigation toc: --> <li><a href="#ex2-hadamard-and-phase-gates" style="font-size: 80%;">Ex2: Hadamard and Phase gates</a></li>
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<!-- navigation toc: --> <li><a href="#ex3-traces-of-operators" style="font-size: 80%;">Ex3: Traces of operators</a></li>
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<!-- navigation toc: --> <li><a href="#ex4-exponentiated-operators" style="font-size: 80%;">Ex4: Exponentiated operators</a></li>
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<!-- navigation toc: --> <li><a href="#ex5-reduced-density-operators-i" style="font-size: 80%;">Ex5: Reduced density operators I</a></li>
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<!-- navigation toc: --> <li><a href="#ex6-reduced-density-operators-ii" style="font-size: 80%;">Ex6: Reduced density operators II</a></li>
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<!-- navigation toc: --> <li><a href="#1-traces-of-operators" style="font-size: 80%;">1: Traces of operators</a></li>
222+
<!-- navigation toc: --> <li><a href="#2-exponentiated-operators" style="font-size: 80%;">2: Exponentiated operators</a></li>
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<!-- navigation toc: --> <li><a href="#3-reduced-density-operators-i" style="font-size: 80%;">3: Reduced density operators I</a></li>
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<!-- navigation toc: --> <li><a href="#4-reduced-density-operators-ii" style="font-size: 80%;">4: Reduced density operators II</a></li>
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<!-- navigation toc: --> <li><a href="#the-next-lecture-february-4-2026" style="font-size: 80%;">The next lecture, February 4, 2026</a></li>
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</ul>
@@ -1146,30 +1131,11 @@ <h2 id="maximally-entangled" class="anchor">Maximally entangled </h2>
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$$
11471132

11481133

1149-
<!-- !split -->
1150-
<h2 id="density-matrices-in-more-detail" class="anchor">Density matrices in more detail </h2>
1151-
11521134
<!-- !split -->
11531135
<h2 id="second-exercise-set" class="anchor">Second exercise set </h2>
11541136

1155-
<p>We bring back the last two exercises from last week as they are meant to build the basis for
1156-
the two projects we will work on during the semester. The first
1157-
project deals with implementing the so-called
1158-
<b>Variational Quantum Eigensolver</b> algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
1159-
</p>
1160-
1161-
<!-- !split -->
1162-
<h2 id="ex1-one-qubit-basis-and-pauli-matrices" class="anchor">Ex1: One-qubit basis and Pauli matrices </h2>
1163-
1164-
<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
1165-
1166-
<!-- !split -->
1167-
<h2 id="ex2-hadamard-and-phase-gates" class="anchor">Ex2: Hadamard and Phase gates </h2>
1168-
1169-
<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
1170-
11711137
<!-- !split -->
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<h2 id="ex3-traces-of-operators" class="anchor">Ex3: Traces of operators </h2>
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<h2 id="1-traces-of-operators" class="anchor">1: Traces of operators </h2>
11731139

11741140
<p>Prove that the trace is cyclic, that is for three operators \( \boldsymbol{A} \), \( \boldsymbol{B} \) and \( \boldsymbol{C} \), we have</p>
11751141
$$
@@ -1178,7 +1144,7 @@ <h2 id="ex3-traces-of-operators" class="anchor">Ex3: Traces of operators </h2>
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11791145

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<!-- !split -->
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<h2 id="ex4-exponentiated-operators" class="anchor">Ex4: Exponentiated operators </h2>
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<h2 id="2-exponentiated-operators" class="anchor">2: Exponentiated operators </h2>
11821148

11831149
<p>Let \( \boldsymbol{A} \) be an operator on a vector space satisfying \( \boldsymbol{A}^2=1 \) and \( \alpha \) any real constant. Show that</p>
11841150
$$
@@ -1188,12 +1154,12 @@ <h2 id="ex4-exponentiated-operators" class="anchor">Ex4: Exponentiated operators
11881154
<p>Does this apply to the Pauli matrices?</p>
11891155

11901156
<!-- !split -->
1191-
<h2 id="ex5-reduced-density-operators-i" class="anchor">Ex5: Reduced density operators I </h2>
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<h2 id="3-reduced-density-operators-i" class="anchor">3: Reduced density operators I </h2>
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11931159
<p>For each of the Bell states, find the reduced density operator/matrix for each qubit.</p>
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11951161
<!-- !split -->
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<h2 id="ex6-reduced-density-operators-ii" class="anchor">Ex6: Reduced density operators II </h2>
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<h2 id="4-reduced-density-operators-ii" class="anchor">4: Reduced density operators II </h2>
11971163

11981164
<p>Suppose we have a composite system which consists of systems \( A \) and
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\( B \) in the state \( \vert a\rangle \otimes \vert b\rangle \), where \( \vert

‎doc/pub/week2/html/week2-reveal.html‎

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@@ -1245,34 +1245,12 @@ <h2 id="maximally-entangled">Maximally entangled </h2>
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="density-matrices-in-more-detail">Density matrices in more detail </h2>
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</section>
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<section>
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<h2 id="second-exercise-set">Second exercise set </h2>
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1255-
<p>We bring back the last two exercises from last week as they are meant to build the basis for
1256-
the two projects we will work on during the semester. The first
1257-
project deals with implementing the so-called
1258-
<b>Variational Quantum Eigensolver</b> algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
1259-
</p>
1260-
</section>
1261-
1262-
<section>
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<h2 id="ex1-one-qubit-basis-and-pauli-matrices">Ex1: One-qubit basis and Pauli matrices </h2>
1264-
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
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</section>
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<section>
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<h2 id="ex2-hadamard-and-phase-gates">Ex2: Hadamard and Phase gates </h2>
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1271-
<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
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</section>
12731251

12741252
<section>
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<h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
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<h2 id="1-traces-of-operators">1: Traces of operators </h2>
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<p>Prove that the trace is cyclic, that is for three operators \( \boldsymbol{A} \), \( \boldsymbol{B} \) and \( \boldsymbol{C} \), we have</p>
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<p>&nbsp;<br>
@@ -1283,7 +1261,7 @@ <h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
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</section>
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12851263
<section>
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<h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
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<h2 id="2-exponentiated-operators">2: Exponentiated operators </h2>
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12881266
<p>Let \( \boldsymbol{A} \) be an operator on a vector space satisfying \( \boldsymbol{A}^2=1 \) and \( \alpha \) any real constant. Show that</p>
12891267
<p>&nbsp;<br>
@@ -1296,13 +1274,13 @@ <h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
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</section>
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<section>
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<h2 id="ex5-reduced-density-operators-i">Ex5: Reduced density operators I </h2>
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<h2 id="3-reduced-density-operators-i">3: Reduced density operators I </h2>
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13011279
<p>For each of the Bell states, find the reduced density operator/matrix for each qubit.</p>
13021280
</section>
13031281

13041282
<section>
1305-
<h2 id="ex6-reduced-density-operators-ii">Ex6: Reduced density operators II </h2>
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<h2 id="4-reduced-density-operators-ii">4: Reduced density operators II </h2>
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13071285
<p>Suppose we have a composite system which consists of systems \( A \) and
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\( B \) in the state \( \vert a\rangle \otimes \vert b\rangle \), where \( \vert

‎doc/pub/week2/html/week2-solarized.html‎

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('Examples of entanglement', 2, None, 'examples-of-entanglement'),
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('Ground state of helium', 2, None, 'ground-state-of-helium'),
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('Maximally entangled', 2, None, 'maximally-entangled'),
147-
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$$
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1084-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="density-matrices-in-more-detail">Density matrices in more detail </h2>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="second-exercise-set">Second exercise set </h2>
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<p>We bring back the last two exercises from last week as they are meant to build the basis for
1091-
the two projects we will work on during the semester. The first
1092-
project deals with implementing the so-called
1093-
<b>Variational Quantum Eigensolver</b> algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
1094-
</p>
1095-
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="ex1-one-qubit-basis-and-pauli-matrices">Ex1: One-qubit basis and Pauli matrices </h2>
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
1100-
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="ex2-hadamard-and-phase-gates">Ex2: Hadamard and Phase gates </h2>
1103-
1104-
<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1107-
<h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
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<h2 id="1-traces-of-operators">1: Traces of operators </h2>
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11091078
<p>Prove that the trace is cyclic, that is for three operators \( \boldsymbol{A} \), \( \boldsymbol{B} \) and \( \boldsymbol{C} \), we have</p>
11101079
$$
@@ -1113,7 +1082,7 @@ <h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
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11141083

11151084
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
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<h2 id="2-exponentiated-operators">2: Exponentiated operators </h2>
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11181087
<p>Let \( \boldsymbol{A} \) be an operator on a vector space satisfying \( \boldsymbol{A}^2=1 \) and \( \alpha \) any real constant. Show that</p>
11191088
$$
@@ -1123,12 +1092,12 @@ <h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
11231092
<p>Does this apply to the Pauli matrices?</p>
11241093

11251094
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1126-
<h2 id="ex5-reduced-density-operators-i">Ex5: Reduced density operators I </h2>
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<h2 id="3-reduced-density-operators-i">3: Reduced density operators I </h2>
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11281097
<p>For each of the Bell states, find the reduced density operator/matrix for each qubit.</p>
11291098

11301099
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1131-
<h2 id="ex6-reduced-density-operators-ii">Ex6: Reduced density operators II </h2>
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<h2 id="4-reduced-density-operators-ii">4: Reduced density operators II </h2>
11321101

11331102
<p>Suppose we have a composite system which consists of systems \( A \) and
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\( B \) in the state \( \vert a\rangle \otimes \vert b\rangle \), where \( \vert

‎doc/pub/week2/html/week2.html‎

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('Examples of entanglement', 2, None, 'examples-of-entanglement'),
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224-
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227-
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229-
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230-
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232-
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233-
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225+
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226+
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234227
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237-
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238-
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229+
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230+
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239231
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240232
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242-
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234+
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246-
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247-
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248-
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250238
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None,
@@ -1158,30 +1146,11 @@ <h2 id="maximally-entangled">Maximally entangled </h2>
11581146
$$
11591147

11601148

1161-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="density-matrices-in-more-detail">Density matrices in more detail </h2>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="second-exercise-set">Second exercise set </h2>
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1167-
<p>We bring back the last two exercises from last week as they are meant to build the basis for
1168-
the two projects we will work on during the semester. The first
1169-
project deals with implementing the so-called
1170-
<b>Variational Quantum Eigensolver</b> algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
1171-
</p>
1172-
1173-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1174-
<h2 id="ex1-one-qubit-basis-and-pauli-matrices">Ex1: One-qubit basis and Pauli matrices </h2>
1175-
1176-
<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
1177-
1178-
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1179-
<h2 id="ex2-hadamard-and-phase-gates">Ex2: Hadamard and Phase gates </h2>
1180-
1181-
<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
1182-
11831152
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1184-
<h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
1153+
<h2 id="1-traces-of-operators">1: Traces of operators </h2>
11851154

11861155
<p>Prove that the trace is cyclic, that is for three operators \( \boldsymbol{A} \), \( \boldsymbol{B} \) and \( \boldsymbol{C} \), we have</p>
11871156
$$
@@ -1190,7 +1159,7 @@ <h2 id="ex3-traces-of-operators">Ex3: Traces of operators </h2>
11901159

11911160

11921161
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1193-
<h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
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<h2 id="2-exponentiated-operators">2: Exponentiated operators </h2>
11941163

11951164
<p>Let \( \boldsymbol{A} \) be an operator on a vector space satisfying \( \boldsymbol{A}^2=1 \) and \( \alpha \) any real constant. Show that</p>
11961165
$$
@@ -1200,12 +1169,12 @@ <h2 id="ex4-exponentiated-operators">Ex4: Exponentiated operators </h2>
12001169
<p>Does this apply to the Pauli matrices?</p>
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12021171
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="ex5-reduced-density-operators-i">Ex5: Reduced density operators I </h2>
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<h2 id="3-reduced-density-operators-i">3: Reduced density operators I </h2>
12041173

12051174
<p>For each of the Bell states, find the reduced density operator/matrix for each qubit.</p>
12061175

12071176
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
1208-
<h2 id="ex6-reduced-density-operators-ii">Ex6: Reduced density operators II </h2>
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<h2 id="4-reduced-density-operators-ii">4: Reduced density operators II </h2>
12091178

12101179
<p>Suppose we have a composite system which consists of systems \( A \) and
12111180
\( B \) in the state \( \vert a\rangle \otimes \vert b\rangle \), where \( \vert

‎doc/pub/week2/ipynb/ipynb-week2-src.tar.gz‎

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