From 4ef247dcb3debe9dc8466a02f30da530cee393c8 Mon Sep 17 00:00:00 2001
From: Stefan Stump <stefan.stump@rwth-aachen.de>
Date: Tue, 29 Apr 2025 12:38:47 +0200
Subject: [PATCH] Added chapter 6

---
 _book/Introduction-to-Quantum-Computing.pdf   | Bin 470638 -> 491088 bytes
 _book/content/compositeSystems.html           | 758 ++++++++++++++++++
 _book/content/introduction.html               |   6 +
 _book/content/observingSystems.html           |   6 +
 _book/content/partialObserving.html           |  10 +
 _book/content/probabilisticSystems.html       |   6 +
 _book/content/quantumSystems.html             |   6 +
 _book/index.html                              |  18 +-
 _book/search.json                             |  48 +-
 _book/sitemap.xml                             |  18 +-
 _quarto.yml                                   |   4 +-
 content/compositeSystems.qmd                  |  46 +-
 ...ion.qmd => quantumCircuitsKetNotation.qmd} |   0
 index.qmd                                     |   8 +-
 14 files changed, 901 insertions(+), 33 deletions(-)
 create mode 100644 _book/content/compositeSystems.html
 rename content/{quantumCircutsKetNotation.qmd => quantumCircuitsKetNotation.qmd} (100%)

diff --git a/_book/Introduction-to-Quantum-Computing.pdf b/_book/Introduction-to-Quantum-Computing.pdf
index 724c9cde94584f801bddf395e91a37e04bfa92be..a921b54a4de8934ac4ecd7a2ea0db87849402fca 100644
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+  <li><a href="#constructing-composite-systems" id="toc-constructing-composite-systems" class="nav-link active" data-scroll-target="#constructing-composite-systems"><span class="header-section-number">6.1</span> Constructing composite systems</a></li>
+  <li><a href="#measuring-composite-systems" id="toc-measuring-composite-systems" class="nav-link" data-scroll-target="#measuring-composite-systems"><span class="header-section-number">6.2</span> Measuring composite systems</a></li>
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+
+<p>So far our probabilistic and quantum systems consist of only one single distribution/state. In the real world, quantum computers often have several different registers (variables).</p>
+<p>In theory, we could use a single very big distribution/state to model multiple qubits. For example a 10 qubit system could be modeled with the classical possibilities <span class="math inline">\(0000000000, 0000000001, \dots, 1111111110, 1111111111\)</span>.</p>
+<p>Unfortunately the vector for these states gets really big, for 10 qubits, the vector would have the dimension of <span class="math inline">\(1024\)</span>. Since this is very inconvenient to write down, we need to look at a different solution. For this, we compose different probabilistic or quantum systems with each other.</p>
+<section id="constructing-composite-systems" class="level2" data-number="6.1">
+<h2 data-number="6.1" class="anchored" data-anchor-id="constructing-composite-systems"><span class="header-section-number">6.1</span> Constructing composite systems</h2>
+<div class="callout callout-style-simple callout-note no-icon">
+<div class="callout-body d-flex">
+<div class="callout-icon-container">
+<i class="callout-icon no-icon"></i>
+</div>
+<div class="callout-body-container">
+<div id="def-composite-systems" class="definition theorem">
+<p><span class="theorem-title"><strong>Definition 6.1 (Composite systems / Tensor product)</strong></span> Given two probabilistic or quantum systems <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> with the possibilities of <span class="math inline">\(A\)</span> given by <span class="math inline">\(1,\dots,N\)</span><a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> and a distribution/state <span class="math inline">\(\mu_a\)</span> and with the possibilities of <span class="math inline">\(B\)</span> given by <span class="math inline">\(1,\dots,M\)</span> and a distribution/state <span class="math inline">\(\mu_b\)</span>, the <em>composite</em> system called <span class="math inline">\(AB\)</span> has the possibilities <span class="math display">\[
+11,12,\dots,1M,21,22,\dots,2M,\dots,N1,N2,\dots,NM
+\]</span> and the distribution/state <span class="math inline">\(ab\)</span> of <span class="math inline">\(AB\)</span> is given by the <em>tensor product</em> <span class="math display">\[
+\mu_{ab}
+= \sqrt{2} \otimes \mu_b
+= \begin{pmatrix} \mu_{a_1} \cdot \mu_b\\ \vdots \\ \mu_{a_N} \cdot \mu_b \end{pmatrix}.
+\]</span> This vector has the size <span class="math inline">\(NM\)</span>.</p>
+</div>
+</div>
+</div>
+</div>
+<p>Note that the definition of combining a probabilistic and a quantum system are the same.</p>
+<div class="callout callout-style-simple callout-tip no-icon callout-titled">
+<div class="callout-header d-flex align-content-center">
+<div class="callout-icon-container">
+<i class="callout-icon no-icon"></i>
+</div>
+<div class="callout-title-container flex-fill">
+Example: Composite system
+</div>
+</div>
+<div class="callout-body-container callout-body">
+<p>Let the distributions for the system <span class="math inline">\(A\)</span> with the possibilities <span class="math inline">\(1,2\)</span> and the system <span class="math inline">\(B\)</span> with the possibilities <span class="math inline">\(A,B,C\)</span> be given by <span class="math display">\[
+\begin{aligned}
+\mu_a &amp;= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix},&amp;
+\mu_b &amp;= \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{\sqrt{3}}{2} \end{pmatrix}.
+\end{aligned}
+\]</span> Then the composite system <span class="math inline">\(AB\)</span> has the possibilities <span class="math inline">\(1A, 1B, 1C, 2A, 2B, 2C\)</span> and the distribution <span class="math display">\[
+\mu_{ab}
+= \mu_a \otimes \mu_b
+= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix} \otimes \begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{pmatrix}
+= \begin{pmatrix} \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \cdot 0 \\ \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \cdot 0 \\ -\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \end{pmatrix}.
+= \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{1}{2} \\ -\frac{1}{2} \\ 0 \\ -\frac{1}{2} \end{pmatrix}.
+\]</span></p>
+</div>
+</div>
+<p>We now need a way to apply operators on these combined systems. For this we can also construct the tensor product of either two probabilistic processes or two unitary transformations by using the tensor product of two matrices.</p>
+<div class="callout callout-style-simple callout-note no-icon">
+<div class="callout-body d-flex">
+<div class="callout-icon-container">
+<i class="callout-icon no-icon"></i>
+</div>
+<div class="callout-body-container">
+<div id="def-composite-systems-matrix" class="definition theorem">
+<p><span class="theorem-title"><strong>Definition 6.2 (Composite matrices / Tensor product)</strong></span> Given two matrices <span class="math inline">\(S\)</span> and <span class="math inline">\(T\)</span> with <span class="math inline">\(S\)</span> of the size <span class="math inline">\(N\times N\)</span> and <span class="math inline">\(T\)</span> of the size <span class="math inline">\(M\times M\)</span>. The tensor product <span class="math inline">\(S\otimes T\)</span> of is given by <span class="math display">\[
+S \otimes T
+= \begin{pmatrix}
+    S_{11}T &amp; \dots &amp; S_{1N} T\\
+    \vdots &amp; \ddots &amp; \vdots\\
+    S_{N1} T &amp; \dots &amp; S_{NN}T
+  \end{pmatrix}.
+\]</span></p>
+</div>
+</div>
+</div>
+</div>
+<p>Overall we can say: If we apply <span class="math inline">\(S\)</span> to the system <span class="math inline">\(A\)</span> and <span class="math inline">\(T\)</span> to the system <span class="math inline">\(B\)</span>, we apply <span class="math inline">\(S\otimes T\)</span> to the composite system <span class="math inline">\(AB\)</span>.</p>
+<p>If the distribution <span class="math inline">\(d_{AB}\)</span> of a given probabilistic system <span class="math inline">\(AB\)</span> can be written as a composite of two distributions <span class="math inline">\(d_A\)</span> and <span class="math inline">\(d_B\)</span>, we know that <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are independent of each other. If we cannot write <span class="math inline">\(d_{AB}\)</span> as two separate distributions, the probabilities are depended on each other.</p>
+<p>If the quantum state <span class="math inline">\(\psi_{AB}\)</span> of a given quantum system <span class="math inline">\(AB\)</span> can be written as a composite of two different quantum states <span class="math inline">\(\psi_A\)</span> and <span class="math inline">\(\psi_B\)</span>, the quantum states of <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are independent of each other. If we can not write <span class="math inline">\(\psi_{AB}\)</span> as a tensor product of two quantum systems, the quantum states depend on each other. We call this <em>entangled</em>.</p>
+<div class="callout callout-style-simple callout-note no-icon">
+<div class="callout-body d-flex">
+<div class="callout-icon-container">
+<i class="callout-icon no-icon"></i>
+</div>
+<div class="callout-body-container">
+<div id="lem-unitary-matrix" class="lemma theorem">
+<p><span class="theorem-title"><strong>Lemma 6.1</strong></span> For two (unitary) matrices <span class="math inline">\(A \in \mathbb{C}^{n\times n}\)</span> and <span class="math inline">\(B \in \mathbb{C}^{m \times m}\)</span> and two vectors (quantum states) <span class="math inline">\(\psi \in \mathbb{C}^n\)</span> and <span class="math inline">\(\phi \in \mathbb{C}^m\)</span>, the following applies <span class="math display">\[
+(A \otimes B)(\psi \otimes \phi) = A\psi \otimes B\phi.
+\]</span></p>
+</div>
+</div>
+</div>
+</div>
+</section>
+<section id="measuring-composite-systems" class="level2" data-number="6.2">
+<h2 data-number="6.2" class="anchored" data-anchor-id="measuring-composite-systems"><span class="header-section-number">6.2</span> Measuring composite systems</h2>
+<p>To perform a (partial) observation or (partial) measurement on a composite system <span class="math inline">\(AB\)</span>, we can compose two separate measurements on the systems <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> similar as we constructed the tensor product.</p>
+<div class="callout callout-style-simple callout-note no-icon">
+<div class="callout-body d-flex">
+<div class="callout-icon-container">
+<i class="callout-icon no-icon"></i>
+</div>
+<div class="callout-body-container">
+<div id="def-composite-systems-measure" class="definition theorem">
+<p><span class="theorem-title"><strong>Definition 6.3 (Composite measurements)</strong></span> Given two systems <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> with possibilities <span class="math inline">\(1,\dots, N\)</span> and <span class="math inline">\(1,\dots, M\)</span> and two partial measurements <span class="math inline">\(M_A\)</span> and <span class="math inline">\(M_B\)</span> on systems <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> with alternatives <span class="math inline">\(A_1,\dots,A_N \subseteq \{1,\dots,N\}\)</span> and <span class="math inline">\(B_1,\dots,B_M \subseteq \{1,\dots,M\}\)</span>. The measurement <span class="math inline">\(M_A \otimes M_B\)</span> on <span class="math inline">\(AB\)</span> is a measurement with the alternatives <span class="math inline">\(C_{11},C_{12},\dots,C_{NM}\)</span> where <span class="math inline">\(C_{ij} = A_i \times B_j\)</span>.</p>
+<p>If we only have a set of alternatives for system <span class="math inline">\(A\)</span>, we can do a measurement <span class="math inline">\(M_A \otimes I\)</span> with alternatives <span class="math inline">\(C_1, \dots ,C_N := A \otimes \{1,\dots, M \}\)</span>.</p>
+</div>
+</div>
+</div>
+</div>
+
+
+</section>
+<section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes">
+<hr>
+<ol>
+<li id="fn1"><p>Note that the possibilities are labels <span class="math inline">\(1,\dots,N\)</span> and do not have be the numbers <span class="math inline">\(1,\dots,N\)</span>. We could also label these e.g.&nbsp;<em>red</em>, <em>green</em> and <em>blue</em> (or any other label).<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+</ol>
+</section>
+
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+<nav class="page-navigation">
+  <div class="nav-page nav-page-previous">
+      <a href="../content/partialObserving.html" class="pagination-link" aria-label="Partial observing and measuring systems">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span>
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+  <div class="nav-page nav-page-next">
+  </div>
+</nav>
+</div> <!-- /content -->
+
+
+
+
+</body></html>
\ No newline at end of file
diff --git a/_book/content/introduction.html b/_book/content/introduction.html
index 73392c0..307de00 100644
--- a/_book/content/introduction.html
+++ b/_book/content/introduction.html
@@ -184,6 +184,12 @@ window.Quarto = {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/observingSystems.html b/_book/content/observingSystems.html
index d14b09d..5525919 100644
--- a/_book/content/observingSystems.html
+++ b/_book/content/observingSystems.html
@@ -184,6 +184,12 @@ window.Quarto = {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/partialObserving.html b/_book/content/partialObserving.html
index 7d04b06..917b5df 100644
--- a/_book/content/partialObserving.html
+++ b/_book/content/partialObserving.html
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 <script src="../site_libs/quarto-search/fuse.min.js"></script>
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
+<link href="../content/compositeSystems.html" rel="next">
 <link href="../content/observingSystems.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js" type="module"></script>
 <script src="../site_libs/quarto-html/tabsets/tabsets.js" type="module"></script>
@@ -183,6 +184,12 @@ window.Quarto = {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
@@ -743,6 +750,9 @@ Example: Partially measuring a quantum system
       </a>          
   </div>
   <div class="nav-page nav-page-next">
+      <a href="../content/compositeSystems.html" class="pagination-link" aria-label="Composite Systems">
+        <span class="nav-page-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span> <i class="bi bi-arrow-right-short"></i>
+      </a>
   </div>
 </nav>
 </div> <!-- /content -->
diff --git a/_book/content/probabilisticSystems.html b/_book/content/probabilisticSystems.html
index c78acbd..c852c68 100644
--- a/_book/content/probabilisticSystems.html
+++ b/_book/content/probabilisticSystems.html
@@ -184,6 +184,12 @@ window.Quarto = {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/quantumSystems.html b/_book/content/quantumSystems.html
index 9d7dfab..0504ce2 100644
--- a/_book/content/quantumSystems.html
+++ b/_book/content/quantumSystems.html
@@ -184,6 +184,12 @@ window.Quarto = {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/index.html b/_book/index.html
index 8974c33..30abe1f 100644
--- a/_book/index.html
+++ b/_book/index.html
@@ -9,7 +9,7 @@
 <meta name="author" content="Jannik Hellenkamp">
 <meta name="author" content="Stefan Stump">
 <meta name="author" content="Dominique Unruh">
-<meta name="dcterms.date" content="2025-04-28">
+<meta name="dcterms.date" content="2025-04-29">
 
 <title>Introduction to Quantum Computing</title>
 <style>
@@ -158,6 +158,12 @@ ul.task-list li input[type="checkbox"] {
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  <span class="menu-text"><span class="chapter-number">5</span>&nbsp; <span class="chapter-title">Partial observing and measuring systems</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./content/compositeSystems.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">6</span>&nbsp; <span class="chapter-title">Composite Systems</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
@@ -203,7 +209,7 @@ ul.task-list li input[type="checkbox"] {
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">April 28, 2025</p>
+      <p class="date">April 29, 2025</p>
     </div>
   </div>
   
@@ -218,7 +224,7 @@ ul.task-list li input[type="checkbox"] {
 <section id="welcome" class="level1 unnumbered">
 <h1 class="unnumbered">Welcome</h1>
 <p>These are the lecture notes for the “Introduction to Quantum Computing” lecture held by Dominique Unruh at RWTH Aachen in the summer term 2025. They should be viewed as an addition to the handwritten notes and the lecture recordings.</p>
-<p>The lecture notes will be updated progressively throughout the semester, following the pace of the lectures. Last year’s lecture notes can be found <a href="https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script">here</a>, but please note that they are incomplete and may differ from this years content.</p>
+<p>The lecture notes will be updated progressively throughout the semester, following the pace of the lectures. Last year’s lecture notes can be found <a href="https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script">here</a>, but note that they are incomplete and may differ from this years content.</p>
 <!--If you prefer a `.pdf` or `.epub` file, there is a download available at the top left corner. Please note that these files are autogenerated and some of the ebook readers have difficulties with the formulas. We are still working on a universal solution. You can also find the source code in the top left corner.-->
 <p>If you prefer a <code>.pdf</code> file, there is a download available at the top left corner. An <code>.epub</code> version is being worked on.</p>
 <!--If you spot an error, please send Stefan Stump an e-mail (stefan.stump\@rwth-aachen.de). If you have a question of understanding, please ask it in the Moodle forum.-->
@@ -226,10 +232,10 @@ ul.task-list li input[type="checkbox"] {
 <section id="changelog" class="level2 unnumbered">
 <h2 class="unnumbered anchored" data-anchor-id="changelog">Changelog</h2>
 <section id="section" class="level4">
-<h4 class="anchored" data-anchor-id="section">28.04.2025</h4>
+<h4 class="anchored" data-anchor-id="section">30.04.2025</h4>
 <ul>
-<li>started the lecture notes for summer 2025</li>
-<li>added chapters 1 to 5 (except 4.3)</li>
+<li>started the lecture notes for the summer term 2025</li>
+<li>added chapters 1 to 6 (except 4.3)</li>
 </ul>
 
 
diff --git a/_book/search.json b/_book/search.json
index 13569aa..3392ee7 100644
--- a/_book/search.json
+++ b/_book/search.json
@@ -4,7 +4,7 @@
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     "section": "",
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+    "text": "Welcome\nThese are the lecture notes for the “Introduction to Quantum Computing” lecture held by Dominique Unruh at RWTH Aachen in the summer term 2025. They should be viewed as an addition to the handwritten notes and the lecture recordings.\nThe lecture notes will be updated progressively throughout the semester, following the pace of the lectures. Last year’s lecture notes can be found here, but note that they are incomplete and may differ from this years content.\nIf you prefer a .pdf file, there is a download available at the top left corner. An .epub version is being worked on.\nThese lecture notes are released under the CC BY-NC 4.0 license, which can be found here.",
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@@ -14,7 +14,7 @@
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     "section": "Changelog",
-    "text": "Changelog\n\n28.04.2025\n\nstarted the lecture notes for summer 2025\nadded chapters 1 to 5 (except 4.3)",
+    "text": "Changelog\n\n30.04.2025\n\nstarted the lecture notes for the summer term 2025\nadded chapters 1 to 6 (except 4.3)",
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     ]
@@ -205,5 +205,49 @@
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       "<span class='chapter-number'>5</span>  <span class='chapter-title'>Partial observing and measuring systems</span>"
     ]
+  },
+  {
+    "objectID": "content/compositeSystems.html",
+    "href": "content/compositeSystems.html",
+    "title": "6  Composite Systems",
+    "section": "",
+    "text": "6.1 Constructing composite systems\nSo far our probabilistic and quantum systems consist of only one single distribution/state. In the real world, quantum computers often have several different registers (variables).\nIn theory, we could use a single very big distribution/state to model multiple qubits. For example a 10 qubit system could be modeled with the classical possibilities \\(0000000000, 0000000001, \\dots, 1111111110, 1111111111\\).\nUnfortunately the vector for these states gets really big, for 10 qubits, the vector would have the dimension of \\(1024\\). Since this is very inconvenient to write down, we need to look at a different solution. For this, we compose different probabilistic or quantum systems with each other.\nNote that the definition of combining a probabilistic and a quantum system are the same.\nWe now need a way to apply operators on these combined systems. For this we can also construct the tensor product of either two probabilistic processes or two unitary transformations by using the tensor product of two matrices.\nOverall we can say: If we apply \\(S\\) to the system \\(A\\) and \\(T\\) to the system \\(B\\), we apply \\(S\\otimes T\\) to the composite system \\(AB\\).\nIf the distribution \\(d_{AB}\\) of a given probabilistic system \\(AB\\) can be written as a composite of two distributions \\(d_A\\) and \\(d_B\\), we know that \\(A\\) and \\(B\\) are independent of each other. If we cannot write \\(d_{AB}\\) as two separate distributions, the probabilities are depended on each other.\nIf the quantum state \\(\\psi_{AB}\\) of a given quantum system \\(AB\\) can be written as a composite of two different quantum states \\(\\psi_A\\) and \\(\\psi_B\\), the quantum states of \\(A\\) and \\(B\\) are independent of each other. If we can not write \\(\\psi_{AB}\\) as a tensor product of two quantum systems, the quantum states depend on each other. We call this entangled.",
+    "crumbs": [
+      "Quantum basics",
+      "<span class='chapter-number'>6</span>  <span class='chapter-title'>Composite Systems</span>"
+    ]
+  },
+  {
+    "objectID": "content/compositeSystems.html#constructing-composite-systems",
+    "href": "content/compositeSystems.html#constructing-composite-systems",
+    "title": "6  Composite Systems",
+    "section": "",
+    "text": "Definition 6.1 (Composite systems / Tensor product) Given two probabilistic or quantum systems \\(A\\) and \\(B\\) with the possibilities of \\(A\\) given by \\(1,\\dots,N\\)1 and a distribution/state \\(\\mu_a\\) and with the possibilities of \\(B\\) given by \\(1,\\dots,M\\) and a distribution/state \\(\\mu_b\\), the composite system called \\(AB\\) has the possibilities \\[\n11,12,\\dots,1M,21,22,\\dots,2M,\\dots,N1,N2,\\dots,NM\n\\] and the distribution/state \\(ab\\) of \\(AB\\) is given by the tensor product \\[\n\\mu_{ab}\n= \\sqrt{2} \\otimes \\mu_b\n= \\begin{pmatrix} \\mu_{a_1} \\cdot \\mu_b\\\\ \\vdots \\\\ \\mu_{a_N} \\cdot \\mu_b \\end{pmatrix}.\n\\] This vector has the size \\(NM\\).\n\n\n\n\n\n\n\n\n\n\n\nExample: Composite system\n\n\n\nLet the distributions for the system \\(A\\) with the possibilities \\(1,2\\) and the system \\(B\\) with the possibilities \\(A,B,C\\) be given by \\[\n\\begin{aligned}\n\\mu_a &= \\begin{pmatrix} \\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\end{pmatrix},&\n\\mu_b &= \\begin{pmatrix} \\frac{1}{2} \\\\ 0 \\\\ \\frac{\\sqrt{3}}{2} \\end{pmatrix}.\n\\end{aligned}\n\\] Then the composite system \\(AB\\) has the possibilities \\(1A, 1B, 1C, 2A, 2B, 2C\\) and the distribution \\[\n\\mu_{ab}\n= \\mu_a \\otimes \\mu_b\n= \\begin{pmatrix} \\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\end{pmatrix} \\otimes \\begin{pmatrix} \\frac{1}{\\sqrt{2}} \\\\ 0 \\\\ \\frac{1}{\\sqrt{2}} \\end{pmatrix}\n= \\begin{pmatrix} \\frac{1}{\\sqrt{2}} \\cdot \\frac{1}{\\sqrt{2}} \\\\ \\frac{1}{\\sqrt{2}} \\cdot 0 \\\\ \\frac{1}{\\sqrt{2}} \\cdot \\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\cdot \\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\cdot 0 \\\\ -\\frac{1}{\\sqrt{2}} \\cdot \\frac{1}{\\sqrt{2}} \\end{pmatrix}.\n= \\begin{pmatrix} \\frac{1}{2} \\\\ 0 \\\\ \\frac{1}{2} \\\\ -\\frac{1}{2} \\\\ 0 \\\\ -\\frac{1}{2} \\end{pmatrix}.\n\\]\n\n\n\n\n\n\n\n\n\n\nDefinition 6.2 (Composite matrices / Tensor product) Given two matrices \\(S\\) and \\(T\\) with \\(S\\) of the size \\(N\\times N\\) and \\(T\\) of the size \\(M\\times M\\). The tensor product \\(S\\otimes T\\) of is given by \\[\nS \\otimes T\n= \\begin{pmatrix}\n    S_{11}T & \\dots & S_{1N} T\\\\\n    \\vdots & \\ddots & \\vdots\\\\\n    S_{N1} T & \\dots & S_{NN}T\n  \\end{pmatrix}.\n\\]\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLemma 6.1 For two (unitary) matrices \\(A \\in \\mathbb{C}^{n\\times n}\\) and \\(B \\in \\mathbb{C}^{m \\times m}\\) and two vectors (quantum states) \\(\\psi \\in \\mathbb{C}^n\\) and \\(\\phi \\in \\mathbb{C}^m\\), the following applies \\[\n(A \\otimes B)(\\psi \\otimes \\phi) = A\\psi \\otimes B\\phi.\n\\]",
+    "crumbs": [
+      "Quantum basics",
+      "<span class='chapter-number'>6</span>  <span class='chapter-title'>Composite Systems</span>"
+    ]
+  },
+  {
+    "objectID": "content/compositeSystems.html#measuring-composite-systems",
+    "href": "content/compositeSystems.html#measuring-composite-systems",
+    "title": "6  Composite Systems",
+    "section": "6.2 Measuring composite systems",
+    "text": "6.2 Measuring composite systems\nTo perform a (partial) observation or (partial) measurement on a composite system \\(AB\\), we can compose two separate measurements on the systems \\(A\\) and \\(B\\) similar as we constructed the tensor product.\n\n\n\n\n\n\n\nDefinition 6.3 (Composite measurements) Given two systems \\(A\\) and \\(B\\) with possibilities \\(1,\\dots, N\\) and \\(1,\\dots, M\\) and two partial measurements \\(M_A\\) and \\(M_B\\) on systems \\(A\\) and \\(B\\) with alternatives \\(A_1,\\dots,A_N \\subseteq \\{1,\\dots,N\\}\\) and \\(B_1,\\dots,B_M \\subseteq \\{1,\\dots,M\\}\\). The measurement \\(M_A \\otimes M_B\\) on \\(AB\\) is a measurement with the alternatives \\(C_{11},C_{12},\\dots,C_{NM}\\) where \\(C_{ij} = A_i \\times B_j\\).\nIf we only have a set of alternatives for system \\(A\\), we can do a measurement \\(M_A \\otimes I\\) with alternatives \\(C_1, \\dots ,C_N := A \\otimes \\{1,\\dots, M \\}\\).",
+    "crumbs": [
+      "Quantum basics",
+      "<span class='chapter-number'>6</span>  <span class='chapter-title'>Composite Systems</span>"
+    ]
+  },
+  {
+    "objectID": "content/compositeSystems.html#footnotes",
+    "href": "content/compositeSystems.html#footnotes",
+    "title": "6  Composite Systems",
+    "section": "",
+    "text": "Note that the possibilities are labels \\(1,\\dots,N\\) and do not have be the numbers \\(1,\\dots,N\\). We could also label these e.g. red, green and blue (or any other label).↩︎",
+    "crumbs": [
+      "Quantum basics",
+      "<span class='chapter-number'>6</span>  <span class='chapter-title'>Composite Systems</span>"
+    ]
   }
 ]
\ No newline at end of file
diff --git a/_book/sitemap.xml b/_book/sitemap.xml
index 374b99d..ae19559 100644
--- a/_book/sitemap.xml
+++ b/_book/sitemap.xml
@@ -2,30 +2,34 @@
 <urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/index.html</loc>
-    <lastmod>2025-04-27T23:33:02.624Z</lastmod>
+    <lastmod>2025-04-28T21:28:08.000Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/introduction.html</loc>
-    <lastmod>2025-04-27T23:05:03.486Z</lastmod>
+    <lastmod>2025-04-28T11:42:14.344Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/probabilisticSystems.html</loc>
-    <lastmod>2025-04-27T23:05:03.487Z</lastmod>
+    <lastmod>2025-04-28T11:42:14.344Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/quantumSystems.html</loc>
-    <lastmod>2025-04-27T23:05:03.487Z</lastmod>
+    <lastmod>2025-04-28T21:22:20.793Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/observingSystems.html</loc>
-    <lastmod>2025-04-27T23:13:16.637Z</lastmod>
+    <lastmod>2025-04-28T11:42:14.344Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/partialObserving.html</loc>
-    <lastmod>2025-04-27T23:48:51.577Z</lastmod>
+    <lastmod>2025-04-28T11:42:14.344Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/compositeSystems.html</loc>
+    <lastmod>2025-04-29T10:37:40.514Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.pdf</loc>
-    <lastmod>2025-04-27T23:49:19.626Z</lastmod>
+    <lastmod>2025-04-29T10:38:10.900Z</lastmod>
   </url>
 </urlset>
diff --git a/_quarto.yml b/_quarto.yml
index 7a77967..984461d 100644
--- a/_quarto.yml
+++ b/_quarto.yml
@@ -25,8 +25,8 @@ book:
         - content/quantumSystems.qmd
         - content/observingSystems.qmd
         - content/partialObserving.qmd
-    #    - content/compositeSystems.qmd
-    #    - content/quantumCircutsKetNotation.qmd
+        - content/compositeSystems.qmd
+    #    - content/quantumCircuitsKetNotation.qmd
     #- part: "Quantum algorithms"
     #  chapters:
     #    - content/bernsteinVazirani.qmd
diff --git a/content/compositeSystems.qmd b/content/compositeSystems.qmd
index d3376d8..a1b7abe 100644
--- a/content/compositeSystems.qmd
+++ b/content/compositeSystems.qmd
@@ -7,21 +7,22 @@ In theory, we could use a single very big distribution/state to model multiple q
 
 Unfortunately the vector for these states gets really big, for 10 qubits, the vector would have the dimension of $1024$. Since this is very inconvenient to write down, we need to look at a different solution. For this, we compose different probabilistic or quantum systems with each other. 
 
-## Constructing composite systems  
 
+## Constructing composite systems
 
 ::: {.callout-note appearance="minimal" icon=false}
 ::: {.definition #def-composite-systems}
 
 ## Composite systems / Tensor product
-Given two probabilistic or quantum systems $A$ and $B$ with the possibilities of $A$ given by $1,\dots,N$[^1] and a distribution/state $a$ and with the possibilities of $B$ given by $1,\dots,M$ and a distribution/state $b$, the *composite* system called $AB$ has the possibilities
+Given two probabilistic or quantum systems $A$ and $B$ with the possibilities of $A$ given by $1,\dots,N$[^1] and a distribution/state $\mu_a$ and with the possibilities of $B$ given by $1,\dots,M$ and a distribution/state $\mu_b$, the *composite* system called $AB$ has the possibilities
 $$
 11,12,\dots,1M,21,22,\dots,2M,\dots,N1,N2,\dots,NM
 $$
-
 and the distribution/state $ab$ of $AB$ is given by the *tensor product*
 $$
-ab= a \otimes b = \begin{pmatrix} a_1 \cdot b\\ \vdots \\ a_N \cdot b \end{pmatrix}
+\mu_{ab}
+= \sqrt{2} \otimes \mu_b
+= \begin{pmatrix} \mu_{a_1} \cdot \mu_b\\ \vdots \\ \mu_{a_N} \cdot \mu_b \end{pmatrix}.
 $$
 This vector has the size $NM$.
 :::
@@ -31,16 +32,22 @@ Note that the definition of combining a probabilistic and a quantum system are t
 
 ::: {.callout-tip icon=false}
 
-## Example: Tensor product 
-Given two vectors $a$ and $b$, with 
+## Example: Composite system
+
+Let the distributions for the system $A$ with the possibilities $1,2$ and the system $B$ with the possibilities $A,B,C$ be given by
 $$
 \begin{aligned}
-a &= \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} & b &= \begin{pmatrix} 10 \\ 100 \end{pmatrix}
+\mu_a &= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix},&
+\mu_b &= \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{\sqrt{3}}{2} \end{pmatrix}.
 \end{aligned}
 $$
-the tensor product of those vectors is given by
+Then the composite system $AB$ has the possibilities $1A, 1B, 1C, 2A, 2B, 2C$ and the distribution
 $$
-a\otimes b = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} \otimes \begin{pmatrix} 10 \\ 100 \end{pmatrix} = \begin{pmatrix} 1\cdot 10 \\ 1 \cdot 100 \\ 2\cdot 10 \\ 2 \cdot 100 \\ 4\cdot 10 \\ 4 \cdot 100 \end{pmatrix} = \begin{pmatrix} 10 \\ 100 \\ 20 \\ 200 \\ 40 \\ 400 \end{pmatrix}
+\mu_{ab}
+= \mu_a \otimes \mu_b
+= \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix} \otimes \begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{pmatrix}
+= \begin{pmatrix} \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \cdot 0 \\ \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \cdot 0 \\ -\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \end{pmatrix}.
+= \begin{pmatrix} \frac{1}{2} \\ 0 \\ \frac{1}{2} \\ -\frac{1}{2} \\ 0 \\ -\frac{1}{2} \end{pmatrix}.
 $$
 
 :::
@@ -53,18 +60,33 @@ We now need a way to apply operators on these combined systems. For this we can
 ## Composite matrices / Tensor product
 Given two matrices $S$ and $T$ with $S$ of the size $N\times N$ and $T$ of the size $M\times M$. The tensor product $S\otimes T$ of is given by
 $$
-S\otimes T = \begin{pmatrix} S_{11}T & \dots & S_{1N} T\\ \vdots & \ddots & \vdots\\ S_{N1} T & \dots & S_{NN}T \end{pmatrix}
+S \otimes T
+= \begin{pmatrix}
+    S_{11}T & \dots & S_{1N} T\\
+    \vdots & \ddots & \vdots\\
+    S_{N1} T & \dots & S_{NN}T
+  \end{pmatrix}.
 $$
 :::
 :::
 
 Overall we can say: If we apply $S$ to the system $A$ and $T$ to the system $B$, we apply $S\otimes T$ to the composite system $AB$.
 
-
-If the distribution $d_{AB}$ of a given probabilistic system $AB$ can be written as a composite of two distributions $d_A$ and $d_B$, we know that $A$ and $B$ are independent of each other. If we can not write $d_{AB}$ as two septate systems, the probabilities are depended on each other. 
+If the distribution $d_{AB}$ of a given probabilistic system $AB$ can be written as a composite of two distributions $d_A$ and $d_B$, we know that $A$ and $B$ are independent of each other. If we cannot write $d_{AB}$ as two separate distributions, the probabilities are depended on each other.
 
 If the quantum state $\psi_{AB}$ of a given quantum system $AB$ can be written as a composite of two different quantum states $\psi_A$ and $\psi_B$, the quantum states of $A$ and $B$ are independent of each other. If we can not write $\psi_{AB}$ as a tensor product of two quantum systems, the quantum states depend on each other. We call this *entangled*.
 
+::: {.callout-note appearance="minimal" icon=false}
+::: {.lemma #lem-unitary-matrix}
+
+For two (unitary) matrices $A \in \mathbb{C}^{n\times n}$ and $B \in \mathbb{C}^{m \times m}$ and two vectors (quantum states) $\psi \in \mathbb{C}^n$ and $\phi \in \mathbb{C}^m$, the following applies
+$$
+(A \otimes B)(\psi \otimes \phi) = A\psi \otimes B\phi.
+$$
+:::
+:::
+
+
 ## Measuring composite systems
 
 To perform a (partial) observation or (partial) measurement on a composite system $AB$, we can compose two separate measurements on the systems $A$ and $B$ similar as we constructed the tensor product. 
diff --git a/content/quantumCircutsKetNotation.qmd b/content/quantumCircuitsKetNotation.qmd
similarity index 100%
rename from content/quantumCircutsKetNotation.qmd
rename to content/quantumCircuitsKetNotation.qmd
diff --git a/index.qmd b/index.qmd
index 67514af..b8d763c 100644
--- a/index.qmd
+++ b/index.qmd
@@ -2,7 +2,7 @@
 
 These are the lecture notes for the "Introduction to Quantum Computing" lecture held by Dominique Unruh at RWTH Aachen in the summer term 2025. They should be viewed as an addition to the handwritten notes and the lecture recordings.
 
-The lecture notes will be updated progressively throughout the semester, following the pace of the lectures. Last year's lecture notes can be found [here](https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script), but please note that they are incomplete and may differ from this years content.
+The lecture notes will be updated progressively throughout the semester, following the pace of the lectures. Last year's lecture notes can be found [here](https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script), but note that they are incomplete and may differ from this years content.
 
 ::: {.content-visible when-format="html:js"}
 <!--If you prefer a `.pdf` or `.epub` file, there is a download available at the top left corner. Please note that these files are autogenerated and some of the ebook readers have difficulties with the formulas. We are still working on a universal solution. You can also find the source code in the top left corner.-->
@@ -15,6 +15,6 @@ These lecture notes are released under the CC BY-NC 4.0 license, which can be fo
 
 ## Changelog {.unnumbered}
 
-#### 28.04.2025
-- started the lecture notes for summer 2025
-- added chapters 1 to 5 (except 4.3)
+#### 30.04.2025
+- started the lecture notes for the summer term 2025
+- added chapters 1 to 6 (except 4.3)
-- 
GitLab