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-    <lastmod>2024-05-28T15:41:23.573Z</lastmod>
-  </url>
-  <url>
-    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/quantumAlgorithms.html</loc>
-    <lastmod>2024-05-28T15:28:23.655Z</lastmod>
-  </url>
-  <url>
-    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/bernsteinVazirani.html</loc>
-    <lastmod>2024-05-28T15:32:36.617Z</lastmod>
-  </url>
-  <url>
-    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/shorsAlgorithm.html</loc>
-    <lastmod>2024-05-31T13:08:52.770Z</lastmod>
-  </url>
-  <url>
-    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.pdf</loc>
-    <lastmod>2024-05-28T15:45:27.879Z</lastmod>
-  </url>
-  <url>
-    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.epub</loc>
-    <lastmod>2024-05-28T15:45:28.828Z</lastmod>
-  </url>
-</urlset>

_quarto.yml

@@ -42,7 +42,7 @@ format:
   pdf:
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     papersize: letter # Necessary for scrreprt!
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+    default-image-extension: pdf
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      text: |
         \usepackage{physics}

_book/index.html → index.html

@@ -8,7 +8,7 @@
 
 <meta name="author" content="Jannik Hellenkamp">
 <meta name="author" content="Dominique Unruh">
-<meta name="dcterms.date" content="2024-05-28">
+<meta name="dcterms.date" content="2024-05-31">
 
 <title>Introduction to Quantum Computing</title>
 <style>
@@ -251,7 +251,7 @@ 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">May 28, 2024</p>
+      <p class="date">May 31, 2024</p>
     </div>
   </div>
   
@@ -271,11 +271,11 @@ ul.task-list li input[type="checkbox"] {
 <p>These lecture notes are released under the CC BY-NC 4.0 license, which can be found <a href="https://creativecommons.org/licenses/by-nc/4.0/">here</a>.</p>
 <section id="changelog" class="level2 unnumbered">
 <h2 class="unnumbered anchored" data-anchor-id="changelog">Changelog</h2>
-<section id="version-0.1.2-xx.05.2024" class="level4">
-<h4 class="anchored" data-anchor-id="version-0.1.2-xx.05.2024">Version 0.1.2 (XX.05.2024)</h4>
+<section id="version-0.1.2-31.05.2024" class="level4">
+<h4 class="anchored" data-anchor-id="version-0.1.2-31.05.2024">Version 0.1.2 (31.05.2024)</h4>
 <ul>
 <li>minor changes to chapter 2</li>
-<li>added chapter 3 and 4</li>
+<li>added chapter 9</li>
 </ul>
 </section>
 <section id="version-0.1.1-16.05.2024" class="level4">

_book/observingSystems.html → observingSystems.html

@@ -69,35 +69,6 @@ ul.task-list li input[type="checkbox"] {
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 </head>
 
@@ -243,15 +214,7 @@ window.Quarto = {
 <div id="quarto-sidebar-glass" class="quarto-sidebar-collapse-item" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item"></div>
 <!-- margin-sidebar -->
     <div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
-        <nav id="TOC" role="doc-toc" class="toc-active">
-    <h2 id="toc-title">Table of contents</h2>
-   
-  <ul>
-  <li><a href="#observing-a-probabilistic-system" id="toc-observing-a-probabilistic-system" class="nav-link active" data-scroll-target="#observing-a-probabilistic-system"><span class="header-section-number">4.1</span> Observing a probabilistic system</a></li>
-  <li><a href="#measuring-a-quantum-system" id="toc-measuring-a-quantum-system" class="nav-link" data-scroll-target="#measuring-a-quantum-system"><span class="header-section-number">4.2</span> Measuring a quantum system</a></li>
-  <li><a href="#elitzurvaidman-bomb-tester" id="toc-elitzurvaidman-bomb-tester" class="nav-link" data-scroll-target="#elitzurvaidman-bomb-tester"><span class="header-section-number">4.3</span> Elitzur–Vaidman bomb tester</a></li>
-  </ul>
-</nav>
+        
     </div>
 <!-- main -->
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@@ -275,95 +238,92 @@ window.Quarto = {
 </header>
 
 
-<p>So far we only talked about the description of a probabilistic and a quantum system. We now look into observing/measuring those systems.</p>
-<section id="observing-a-probabilistic-system" class="level2" data-number="4.1">
-<h2 data-number="4.1" class="anchored" data-anchor-id="observing-a-probabilistic-system"><span class="header-section-number">4.1</span> Observing a probabilistic system</h2>
-<p>Observing a probabilistic system is the process of obtaining an outcome from a probability distribution.</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-observing-probabilistic" class="definition theorem">
-<p><span class="theorem-title"><strong>Definition 4.1 (Observing a probabilistic system)</strong></span> Given a probability distribution <span class="math inline">\(d \in \mathbb{R}^n\)</span>, we will get the outcome <span class="math inline">\(i\)</span> with a probability <span class="math inline">\(d_i\)</span>. The new distribution is then <span class="math display">\[
+<!--
+So far we only talked about the description of a probabilistic and a quantum system. We now look into observing/measuring those systems. 
+
+## Observing a probabilistic system 
+
+Observing a probabilistic system is the process of obtaining an outcome from a probability distribution.
+
+::: {.callout-note appearance="minimal" icon=false}
+::: {.definition #def-observing-probabilistic}
+
+## Observing a probabilistic system
+Given a probability distribution $d \in \mathbb{R}^n$, we will get the outcome $i$ with a probability $d_i$. The new distribution is then 
+$$
 e_i = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}  \leftarrow \text{1 at the $i$-th position}
-\]</span></p>
-</div>
-</div>
-</div>
-</div>
-<p>When observing a probabilistic system, the observation is just a passive process with no impact on the system. This means, that there is no difference to the end result, if we observe during the process. We take a look at an example to further understand this.</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: Random 1-bit number
-</div>
-</div>
-<div class="callout-body-container callout-body">
-<p>We usa a the random 1-bit number example to the random 2-bit example from <a href="probabilisticSystems.html" class="quarto-xref"><span>Chapter 2</span></a>. We have a distribution <span class="math inline">\(d_{\text{1-bit}} = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix}\)</span> which represents the probability distribution of generating a 1-bit number with equal probability. We also have a process <span class="math inline">\(A_{\text{flip}} = \begin{pmatrix} \frac{2}{3} &amp;  \frac{1}{3} \\ \frac{1}{3} &amp;  \frac{2}{3} \end{pmatrix}\)</span> which flips the bit with a probability of <span class="math inline">\(\frac{1}{3}\)</span>.</p>
-<p>We look at two different cases: For the first case, we observe only the final distribution and for the second case we observe after the generation of the 1-bit number and we also observe the final distribution.</p>
-<section id="observing-the-final-distribution" class="level5 unnumbered">
-<h5 class="unnumbered anchored" data-anchor-id="observing-the-final-distribution">Observing the final distribution</h5>
-<p>From <a href="probabilisticSystems.html#sec-prob-apply" class="quarto-xref"><span>Section 2.3</span></a> we know that the final distribution <span class="math inline">\(d\)</span> is <span class="math display">\[
-d = A_{\text{flip}} \cdot d_{\text{1-bit}} = \begin{pmatrix} \frac{2}{3} &amp;  \frac{1}{3} \\ \frac{1}{3} &amp;  \frac{2}{3} \end{pmatrix} \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix}
-\]</span> We observe this distribution and will get outcome <span class="math inline">\(0\)</span> and the new distribution <span class="math inline">\(d = e_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\)</span> with a probability of <span class="math inline">\(d_0 = \frac{1}{2}\)</span> and the outcome <span class="math inline">\(1\)</span> and the new distribution <span class="math inline">\(d = e_1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)</span> with a probability of <span class="math inline">\(d_1 = \frac{1}{2}\)</span>.</p>
-</section>
-<section id="observing-after-generation" class="level5 unnumbered">
-<h5 class="unnumbered anchored" data-anchor-id="observing-after-generation">Observing after generation</h5>
-<p>We now observe the system after the generation of the 1-bit number and also observe the final distribution</p>
-</section>
-</div>
-</div>
-</section>
-<section id="measuring-a-quantum-system" class="level2" data-number="4.2">
-<h2 data-number="4.2" class="anchored" data-anchor-id="measuring-a-quantum-system"><span class="header-section-number">4.2</span> Measuring a quantum system</h2>
-<p>Unlike in the probabilistic system, the “observation” of a quantum system is called <em>measuring</em>. The definition is similar to the observation of a probabilistic system, except that we need to take the absolute square of the amplitude to get the probability and that the state after measuring is called <em>post measurement state</em> (p.m.s.).</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-observing-probabilistic" class="definition theorem">
-<p><span class="theorem-title"><strong>Definition 4.2 (Measuring a quantum system)</strong></span> Given a quantum State <span class="math inline">\(\psi \in \mathbb{C}^n\)</span>, we will get the outcome <span class="math inline">\(i\)</span> with a probability <span class="math inline">\(|\psi_i|^2\)</span>. The post measurement state (p.m.s.) is then <span class="math display">\[
+$$
+:::
+:::
+When observing a probabilistic system, the observation is just a passive process with no impact on the system. This means, that there is no difference to the end result, if we observe during the process. We take a look at an example to further understand this. 
+
+::: {.callout-tip icon=false}
+
+## Example: Random 1-bit number
+
+We usa a the random 1-bit number example to the random 2-bit example from @sec-prob.
+We have a distribution $d_{\text{1-bit}} = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix}$ which represents the probability distribution of generating a 1-bit number with equal probability. We also have a process $A_{\text{flip}} = \begin{pmatrix} \frac{2}{3} &  \frac{1}{3} \\ \frac{1}{3} &  \frac{2}{3} \end{pmatrix}$ which flips the bit with a probability of $\frac{1}{3}$. 
+
+We look at two different cases: For the first case, we observe only the final distribution and for the second case we observe after the generation of the 1-bit number and we also observe the final distribution. 
+
+##### Observing the final distribution {.unnumbered}
+
+From @sec-prob-apply we know that the final distribution $d$ is 
+$$
+d = A_{\text{flip}} \cdot d_{\text{1-bit}} = \begin{pmatrix} \frac{2}{3} &  \frac{1}{3} \\ \frac{1}{3} &  \frac{2}{3} \end{pmatrix} \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{2} \end{pmatrix}
+$$
+We observe this distribution and will get outcome $0$ and the new distribution $d = e_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ with a probability of $d_0 = \frac{1}{2}$ and the outcome $1$ and the new distribution $d = e_1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ with a probability of $d_1 = \frac{1}{2}$.
+
+##### Observing after generation {.unnumbered}
+
+We now observe the system after the generation of the 1-bit number and also observe the final distribution
+
+:::
+
+## Measuring a quantum system
+
+Unlike in the probabilistic system, the "observation" of a quantum system is called *measuring*. The definition is similar to the observation of a probabilistic system, except that we need to take the absolute square of the amplitude to get the probability and that the state after measuring is called *post measurement state* (p.m.s.). 
+
+::: {.callout-note appearance="minimal" icon=false}
+::: {.definition #def-observing-probabilistic}
+
+## Measuring a quantum system
+Given a quantum State $\psi \in \mathbb{C}^n$, we will get the outcome $i$ with a probability $|\psi_i|^2$. The post measurement state (p.m.s.) is then 
+$$
 e_i = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}  \leftarrow \text{1 at the $i$-th position}
-\]</span></p>
-</div>
-</div>
-</div>
-</div>
-<p>With this similarity in to the probabilistic observation in the definition, one might assume, that measuring a quantum state has also no impact on the system. <strong>This is not the case, measuring a quantum state changes the system!</strong> We can see this effect with an example:</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: Measuring a quantum system
-</div>
-</div>
-<div class="callout-body-container callout-body">
-<p>Let <span class="math inline">\(\psi = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}\)</span> be a quantum state and <span class="math inline">\(H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; -1 \end{pmatrix}\)</span> be a unitary transformation. We look at two different cases: First we apply <span class="math inline">\(H\)</span> immediately and then measure the system. As a second case, we do a measurement before the application of the <span class="math inline">\(H\)</span> unitary and then a measurement after applying it.</p>
-<section id="measure-the-final-state" class="level5 unnumbered">
-<h5 class="unnumbered anchored" data-anchor-id="measure-the-final-state">Measure the final state</h5>
-<p>We first calculate the state after applying <span class="math inline">\(H\)</span>: <span class="math display">\[
-H\psi = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; -1 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix} =  \begin{pmatrix} 1 \\ 0 \end{pmatrix}
-\]</span></p>
-<p>Measuring this state will get the outcome <span class="math inline">\(0\)</span> with probability <span class="math inline">\(|\psi_0|^2 = 1\)</span> and have the post measurement state <span class="math inline">\(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\)</span>.</p>
-</section>
-</div>
-</div>
-</section>
-<section id="elitzurvaidman-bomb-tester" class="level2" data-number="4.3">
-<h2 data-number="4.3" class="anchored" data-anchor-id="elitzurvaidman-bomb-tester"><span class="header-section-number">4.3</span> Elitzur–Vaidman bomb tester</h2>
-<p>This section will be updated later on.</p>
+$$
+:::
+:::
+
+With this similarity in to the probabilistic observation in the definition, one might assume, that measuring a quantum state has also no impact on the system. **This is not the case, measuring a quantum state changes the system!** We can see this effect with an example:
+
+::: {.callout-tip icon=false}
+
+## Example: Measuring a quantum system
+
+Let $\psi = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$ be a quantum state and $H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ be a unitary transformation. We look at two different cases: First we apply $H$ immediately and then measure the system. As a second case, we do a measurement before the application of the $H$ unitary and then a measurement after applying it. 
+
+##### Measure the final state {.unnumbered}
+
+We first calculate the state after applying $H$:
+$$
+H\psi = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix} =  \begin{pmatrix} 1 \\ 0 \end{pmatrix}
+$$
+
+Measuring this state will get the outcome $0$ with probability $|\psi_0|^2 = 1$ and have the post measurement state $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
+
+
+:::
+
+
+
+## Elitzur–Vaidman bomb tester
+
+This section will be updated later on. 
+
+-->
 
 
-</section>
 
 </main> <!-- /main -->
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 # Observing probabilistic and measuring quantum systems
 
+<!--
 So far we only talked about the description of a probabilistic and a quantum system. We now look into observing/measuring those systems. 
 
 ## Observing a probabilistic system 
@@ -80,4 +81,6 @@ Measuring this state will get the outcome $0$ with probability $|\psi_0|^2 = 1$
 
 ## Elitzur–Vaidman bomb tester
 
-This section will be updated later on. 
\ No newline at end of file
+This section will be updated later on. 
+
+-->
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