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diff --git a/_book/probabilisticSystems.html b/_book/probabilisticSystems.html
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--- a/_book/probabilisticSystems.html
+++ b/_book/probabilisticSystems.html
@@ -193,7 +193,7 @@ window.Quarto = {
<h2 id="toc-title">Table of contents</h2>
<ul>
- <li><a href="#deterministic-classical-possibilities" id="toc-deterministic-classical-possibilities" class="nav-link active" data-scroll-target="#deterministic-classical-possibilities"><span class="header-section-number">2.1</span> Deterministic (classical) possibilities</a></li>
+ <li><a href="#deterministic-possibilities" id="toc-deterministic-possibilities" class="nav-link active" data-scroll-target="#deterministic-possibilities"><span class="header-section-number">2.1</span> Deterministic possibilities</a></li>
<li><a href="#probability-distribution" id="toc-probability-distribution" class="nav-link" data-scroll-target="#probability-distribution"><span class="header-section-number">2.2</span> Probability distribution</a></li>
<li><a href="#probabilistic-processes" id="toc-probabilistic-processes" class="nav-link" data-scroll-target="#probabilistic-processes"><span class="header-section-number">2.3</span> Probabilistic processes</a>
<ul class="collapse">
@@ -225,9 +225,9 @@ window.Quarto = {
<p>To describe a quantum computer mathematically, we can do math similar to the known topic of probabilistic systems. We therefore first look into describing a probabilistic system.</p>
-<section id="deterministic-classical-possibilities" class="level2" data-number="2.1">
-<h2 data-number="2.1" class="anchored" data-anchor-id="deterministic-classical-possibilities"><span class="header-section-number">2.1</span> Deterministic (classical) possibilities</h2>
-<p>At first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be <em>heads</em> or <em>tails</em> and for a dice this could be the labels of the different sides. We call these possibilities <em>classical possibilities</em>. Note that we will only be using a <em>finite</em> number of possibilities.</p>
+<section id="deterministic-possibilities" class="level2" data-number="2.1">
+<h2 data-number="2.1" class="anchored" data-anchor-id="deterministic-possibilities"><span class="header-section-number">2.1</span> Deterministic possibilities</h2>
+<p>At first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be <em>heads</em> or <em>tails</em> and for a dice this could be the labels of the different sides. We call these possibilities <em>deterministic possibilities</em>. Note that we will only be using a <em>finite</em> number of possibilities.</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">
@@ -238,13 +238,13 @@ Example: Random 2-bit number
</div>
</div>
<div class="callout-body-container callout-body">
-<p>Imagine you have a random number generator, which outputs 2-bit numbers. The classical possibilities of this generator are <span class="math inline">\(00\)</span>, <span class="math inline">\(01\)</span>, <span class="math inline">\(10\)</span> and <span class="math inline">\(11\)</span>.</p>
+<p>Imagine you have a random number generator, which outputs 2-bit numbers. The deterministic possibilities of this generator are <span class="math inline">\(00\)</span>, <span class="math inline">\(01\)</span>, <span class="math inline">\(10\)</span> and <span class="math inline">\(11\)</span>.</p>
</div>
</div>
</section>
<section id="probability-distribution" class="level2" data-number="2.2">
<h2 data-number="2.2" class="anchored" data-anchor-id="probability-distribution"><span class="header-section-number">2.2</span> Probability distribution</h2>
-<p>Next, we need to assign each possibility a probability. We write this as <span class="math inline">\(\Pr[x]=p\)</span> where <span class="math inline">\(p \in [0,1]\)</span> is the probability of the classical possibility <span class="math inline">\(x\)</span>.</p>
+<p>Next, we need to assign each possibility a probability. We write this as <span class="math inline">\(\Pr[x]=p\)</span> where <span class="math inline">\(p \in [0,1]\)</span> is the probability of the deterministic possibility <span class="math inline">\(x\)</span>.</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">
@@ -271,7 +271,7 @@ Example: Coin flip
</div>
</div>
</div>
-<p>This vector has <span class="math inline">\(n\)</span> entries, where each entry corresponds to a classical possibility <span class="math inline">\(X\)</span> and the probability of <span class="math inline">\(X\)</span> is <span class="math inline">\(\Pr[X] = d_i\)</span>. The sum over all probabilities has to be <span class="math inline">\(1\)</span> and each entry needs to be nonnegative in order to be a valid probability.</p>
+<p>This vector has <span class="math inline">\(n\)</span> entries, where each entry corresponds to a deterministic possibility <span class="math inline">\(X\)</span> and the probability of <span class="math inline">\(X\)</span> is <span class="math inline">\(\Pr[X] = d_i\)</span>. The sum over all probabilities has to be <span class="math inline">\(1\)</span> and each entry needs to be nonnegative in order to be a valid probability.</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">
@@ -295,7 +295,7 @@ Example (continued): Random 2-bit number
</div>
</div>
<div class="callout-body-container callout-body">
-<p>Recall your random 2-bit number generator from above. Imagine your generator outputs each classical possibility with equal probability, except for the possibility <span class="math inline">\(00\)</span>, which is never generated. The corresponding probability distribution would be <span class="math display">\[
+<p>Recall your random 2-bit number generator from above. Imagine your generator outputs each deterministic possibility with equal probability, except for the possibility <span class="math inline">\(00\)</span>, which is never generated. The corresponding probability distribution would be <span class="math display">\[
d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix}
\]</span></p>
</div>
@@ -304,7 +304,7 @@ d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3}
<section id="probabilistic-processes" class="level2" data-number="2.3">
<h2 data-number="2.3" class="anchored" data-anchor-id="probabilistic-processes"><span class="header-section-number">2.3</span> Probabilistic processes</h2>
<p>With a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a <em>probabilistic process</em>.</p>
-<p>A probabilistic process is a collection of <span class="math inline">\(n\)</span> probability distributions, where for each classical possibility <span class="math inline">\(i\)</span> there is a probability distribution <span class="math inline">\(a_i\)</span>. This means, that if the system is in classical possibility <span class="math inline">\(i\)</span> before the process is applied, the system will afterwards be distributed according to <span class="math inline">\(a_i\)</span>. We can write this as a matrix, where each column is a probability distribution <span class="math inline">\(a_i\)</span>.</p>
+<p>A probabilistic process is a collection of <span class="math inline">\(n\)</span> probability distributions, where for each deterministic possibility <span class="math inline">\(i\)</span> there is a probability distribution <span class="math inline">\(a_i\)</span>. This means, that if the system is in deterministic possibility <span class="math inline">\(i\)</span> before the process is applied, the system will afterwards be distributed according to <span class="math inline">\(a_i\)</span>. We can write this as a matrix, where each column is a probability distribution <span class="math inline">\(a_i\)</span>.</p>
<div class="callout callout-style-simple callout-note no-icon">
<div class="callout-body d-flex">
<div class="callout-icon-container">
@@ -328,7 +328,7 @@ Example (continued): Random 2-bit number
</div>
</div>
<div class="callout-body-container callout-body">
-<p>Imagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of <span class="math inline">\(\frac{1}{3}\)</span>. The probability distributions for each of the classical possibility would then be <span class="math display">\[
+<p>Imagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of <span class="math inline">\(\frac{1}{3}\)</span>. The probability distributions for each of the deterministic possibility would then be <span class="math display">\[
a_{00} = \begin{pmatrix} \frac{2}{3} \\ 0 \\ 0 \\ \frac{1}{3} \end{pmatrix}, a_{01} =\begin{pmatrix} 0 \\ \frac{2}{3} \\ \frac{1}{3} \\ 0 \end{pmatrix}, a_{10} =\begin{pmatrix} 0 \\ \frac{1}{3} \\ \frac{2}{3} \\ 0 \end{pmatrix}, a_{11} = \begin{pmatrix} \frac{1}{3} \\ 0 \\ 0 \\ \frac{2}{3} \end{pmatrix}
\]</span> From this we can construct the process as a matrix from these processes as follows: <span class="math display">\[
A_{\text{flip}} = \begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \end{pmatrix} = \begin{pmatrix} \frac{2}{3} & 0 & 0 & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3} & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 \\ \frac{1}{3} & 0 & 0 & \frac{2}{3} \end{pmatrix}
diff --git a/_book/search.json b/_book/search.json
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@@ -46,18 +46,18 @@
"href": "probabilisticSystems.html",
"title": "2 Probabilistic systems",
"section": "",
- "text": "2.1 Deterministic (classical) possibilities\nAt first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be heads or tails and for a dice this could be the labels of the different sides. We call these possibilities classical possibilities. Note that we will only be using a finite number of possibilities.",
+ "text": "2.1 Deterministic possibilities\nAt first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be heads or tails and for a dice this could be the labels of the different sides. We call these possibilities deterministic possibilities. Note that we will only be using a finite number of possibilities.",
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- "text": "Example: Random 2-bit number\n\n\n\nImagine you have a random number generator, which outputs 2-bit numbers. The classical possibilities of this generator are \\(00\\), \\(01\\), \\(10\\) and \\(11\\).",
+ "text": "Example: Random 2-bit number\n\n\n\nImagine you have a random number generator, which outputs 2-bit numbers. The deterministic possibilities of this generator are \\(00\\), \\(01\\), \\(10\\) and \\(11\\).",
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"href": "probabilisticSystems.html#probability-distribution",
"title": "2 Probabilistic systems",
"section": "2.2 Probability distribution",
- "text": "2.2 Probability distribution\nNext, we need to assign each possibility a probability. We write this as \\(\\Pr[x]=p\\) where \\(p \\in [0,1]\\) is the probability of the classical possibility \\(x\\).\n\n\n\n\n\n\nExample: Coin flip\n\n\n\nFor a coin flip the probability of heads would be \\(\\Pr[\\text{heads}] = \\frac{1}{2}\\) and the probability for tails would be \\(\\Pr[\\text{tails}] = \\frac{1}{2}\\).\n\n\nIf we combine all probabilities for all the possible outcomes and write them as a vector, we get a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.1 (Probability distribution) A vector \\(d \\in \\mathbb{R}^n\\) is a valid probability distribution iff \\(\\sum d_i = 1\\) and \\(\\forall i\\) \\(d_i \\geq 0\\)\n\n\n\n\nThis vector has \\(n\\) entries, where each entry corresponds to a classical possibility \\(X\\) and the probability of \\(X\\) is \\(\\Pr[X] = d_i\\). The sum over all probabilities has to be \\(1\\) and each entry needs to be nonnegative in order to be a valid probability.\n\n\n\n\n\n\nExample (continued): Coin flip\n\n\n\nFor a coin flip the probability distribution would be \\(d_{\\text{coin}} \\in \\mathbb{R}^2\\) with \\(d = \\begin{pmatrix}\\frac{1}{2}\\\\ \\frac{1}{2} \\end{pmatrix}\\)\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall your random 2-bit number generator from above. Imagine your generator outputs each classical possibility with equal probability, except for the possibility \\(00\\), which is never generated. The corresponding probability distribution would be \\[\nd_{\\text{2-bit}} = \\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix}\n\\]",
+ "text": "2.2 Probability distribution\nNext, we need to assign each possibility a probability. We write this as \\(\\Pr[x]=p\\) where \\(p \\in [0,1]\\) is the probability of the deterministic possibility \\(x\\).\n\n\n\n\n\n\nExample: Coin flip\n\n\n\nFor a coin flip the probability of heads would be \\(\\Pr[\\text{heads}] = \\frac{1}{2}\\) and the probability for tails would be \\(\\Pr[\\text{tails}] = \\frac{1}{2}\\).\n\n\nIf we combine all probabilities for all the possible outcomes and write them as a vector, we get a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.1 (Probability distribution) A vector \\(d \\in \\mathbb{R}^n\\) is a valid probability distribution iff \\(\\sum d_i = 1\\) and \\(\\forall i\\) \\(d_i \\geq 0\\)\n\n\n\n\nThis vector has \\(n\\) entries, where each entry corresponds to a deterministic possibility \\(X\\) and the probability of \\(X\\) is \\(\\Pr[X] = d_i\\). The sum over all probabilities has to be \\(1\\) and each entry needs to be nonnegative in order to be a valid probability.\n\n\n\n\n\n\nExample (continued): Coin flip\n\n\n\nFor a coin flip the probability distribution would be \\(d_{\\text{coin}} \\in \\mathbb{R}^2\\) with \\(d = \\begin{pmatrix}\\frac{1}{2}\\\\ \\frac{1}{2} \\end{pmatrix}\\)\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall your random 2-bit number generator from above. Imagine your generator outputs each deterministic possibility with equal probability, except for the possibility \\(00\\), which is never generated. The corresponding probability distribution would be \\[\nd_{\\text{2-bit}} = \\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix}\n\\]",
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"href": "probabilisticSystems.html#probabilistic-processes",
"title": "2 Probabilistic systems",
"section": "2.3 Probabilistic processes",
- "text": "2.3 Probabilistic processes\nWith a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a probabilistic process.\nA probabilistic process is a collection of \\(n\\) probability distributions, where for each classical possibility \\(i\\) there is a probability distribution \\(a_i\\). This means, that if the system is in classical possibility \\(i\\) before the process is applied, the system will afterwards be distributed according to \\(a_i\\). We can write this as a matrix, where each column is a probability distribution \\(a_i\\).\n\n\n\n\n\n\n\nDefinition 2.2 (Probabilistic process) A matrix \\(A \\in \\mathbb{R}^{n\\times n}\\) is a valid probabilistic process iff for every column \\(a\\) of \\(A\\), \\(a\\) is a valid probability distribution.\n\n\n\n\nFrom Definition 2.1 we know that a valid probability distribution \\(a\\) has the properties \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\), therefore a matrix \\(A\\) is a probabilistic process iff \\(A \\in \\mathbb{R}^{n \\times n}\\) with \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\) . Such a matrix is also called a stochastic matrix.\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nImagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of \\(\\frac{1}{3}\\). The probability distributions for each of the classical possibility would then be \\[\na_{00} = \\begin{pmatrix} \\frac{2}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{1}{3} \\end{pmatrix}, a_{01} =\\begin{pmatrix} 0 \\\\ \\frac{2}{3} \\\\ \\frac{1}{3} \\\\ 0 \\end{pmatrix}, a_{10} =\\begin{pmatrix} 0 \\\\ \\frac{1}{3} \\\\ \\frac{2}{3} \\\\ 0 \\end{pmatrix}, a_{11} = \\begin{pmatrix} \\frac{1}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{2}{3} \\end{pmatrix}\n\\] From this we can construct the process as a matrix from these processes as follows: \\[\nA_{\\text{flip}} = \\begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \\end{pmatrix} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 & \\frac{1}{3} \\\\ 0 & \\frac{2}{3} & \\frac{1}{3} & 0 \\\\ 0 & \\frac{1}{3} & \\frac{2}{3} & 0 \\\\ \\frac{1}{3} & 0 & 0 & \\frac{2}{3} \\end{pmatrix}\n\\]\n\n\n\nApplying a probabilistic process\nHaving defined probability distributions and probabilistic processes, we can now combine these two elements and apply a probabilistic process on a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.3 (Applying a probabilistic process) Given an initial probability distribution \\(x\\) and a probabilistic process \\(A\\), the result \\(y\\) of applying the process \\(A\\) is defined as \\[\ny = Ax\n\\]\n\n\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall the 2-bit number generator and the bit flip from above. Imagine you would first draw a random 2-bit number from the generator and then run the bit flip device. We already know that the probability distribution of the generator is \\(M_\\text{2-bit}\\). Using \\(A_\\text{flip}\\) we can calculate the final probability distribution: \\[\nA_\\text{flip} \\cdot d_\\text{2-bit} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 & \\frac{1}{3} \\\\ 0 & \\frac{2}{3} & \\frac{1}{3} & 0 \\\\ 0 & \\frac{1}{3} & \\frac{2}{3} & 0 \\\\ \\frac{1}{3} & 0 & 0 & \\frac{2}{3} \\end{pmatrix}\\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix} = \\begin{pmatrix} \\frac{1}{9} \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{2}{9} \\end{pmatrix}\n\\]",
+ "text": "2.3 Probabilistic processes\nWith a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a probabilistic process.\nA probabilistic process is a collection of \\(n\\) probability distributions, where for each deterministic possibility \\(i\\) there is a probability distribution \\(a_i\\). This means, that if the system is in deterministic possibility \\(i\\) before the process is applied, the system will afterwards be distributed according to \\(a_i\\). We can write this as a matrix, where each column is a probability distribution \\(a_i\\).\n\n\n\n\n\n\n\nDefinition 2.2 (Probabilistic process) A matrix \\(A \\in \\mathbb{R}^{n\\times n}\\) is a valid probabilistic process iff for every column \\(a\\) of \\(A\\), \\(a\\) is a valid probability distribution.\n\n\n\n\nFrom Definition 2.1 we know that a valid probability distribution \\(a\\) has the properties \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\), therefore a matrix \\(A\\) is a probabilistic process iff \\(A \\in \\mathbb{R}^{n \\times n}\\) with \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\) . Such a matrix is also called a stochastic matrix.\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nImagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of \\(\\frac{1}{3}\\). The probability distributions for each of the deterministic possibility would then be \\[\na_{00} = \\begin{pmatrix} \\frac{2}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{1}{3} \\end{pmatrix}, a_{01} =\\begin{pmatrix} 0 \\\\ \\frac{2}{3} \\\\ \\frac{1}{3} \\\\ 0 \\end{pmatrix}, a_{10} =\\begin{pmatrix} 0 \\\\ \\frac{1}{3} \\\\ \\frac{2}{3} \\\\ 0 \\end{pmatrix}, a_{11} = \\begin{pmatrix} \\frac{1}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{2}{3} \\end{pmatrix}\n\\] From this we can construct the process as a matrix from these processes as follows: \\[\nA_{\\text{flip}} = \\begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \\end{pmatrix} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 & \\frac{1}{3} \\\\ 0 & \\frac{2}{3} & \\frac{1}{3} & 0 \\\\ 0 & \\frac{1}{3} & \\frac{2}{3} & 0 \\\\ \\frac{1}{3} & 0 & 0 & \\frac{2}{3} \\end{pmatrix}\n\\]\n\n\n\nApplying a probabilistic process\nHaving defined probability distributions and probabilistic processes, we can now combine these two elements and apply a probabilistic process on a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.3 (Applying a probabilistic process) Given an initial probability distribution \\(x\\) and a probabilistic process \\(A\\), the result \\(y\\) of applying the process \\(A\\) is defined as \\[\ny = Ax\n\\]\n\n\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall the 2-bit number generator and the bit flip from above. Imagine you would first draw a random 2-bit number from the generator and then run the bit flip device. We already know that the probability distribution of the generator is \\(M_\\text{2-bit}\\). Using \\(A_\\text{flip}\\) we can calculate the final probability distribution: \\[\nA_\\text{flip} \\cdot d_\\text{2-bit} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 & \\frac{1}{3} \\\\ 0 & \\frac{2}{3} & \\frac{1}{3} & 0 \\\\ 0 & \\frac{1}{3} & \\frac{2}{3} & 0 \\\\ \\frac{1}{3} & 0 & 0 & \\frac{2}{3} \\end{pmatrix}\\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix} = \\begin{pmatrix} \\frac{1}{9} \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{2}{9} \\end{pmatrix}\n\\]",
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diff --git a/probabilisticSystems.qmd b/probabilisticSystems.qmd
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--- a/probabilisticSystems.qmd
+++ b/probabilisticSystems.qmd
@@ -1,17 +1,17 @@
# Probabilistic systems
To describe a quantum computer mathematically, we can do math similar to the known topic of probabilistic systems. We therefore first look into describing a probabilistic system.
-## Deterministic (classical) possibilities
-At first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be *heads* or *tails* and for a dice this could be the labels of the different sides. We call these possibilities *classical possibilities*. Note that we will only be using a *finite* number of possibilities.
+## Deterministic possibilities
+At first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be *heads* or *tails* and for a dice this could be the labels of the different sides. We call these possibilities *deterministic possibilities*. Note that we will only be using a *finite* number of possibilities.
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## Example: Random 2-bit number
-Imagine you have a random number generator, which outputs 2-bit numbers. The classical possibilities of this generator are $00$, $01$, $10$ and $11$.
+Imagine you have a random number generator, which outputs 2-bit numbers. The deterministic possibilities of this generator are $00$, $01$, $10$ and $11$.
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## Probability distribution
-Next, we need to assign each possibility a probability. We write this as $\Pr[x]=p$ where $p \in [0,1]$ is the probability of the classical possibility $x$.
+Next, we need to assign each possibility a probability. We write this as $\Pr[x]=p$ where $p \in [0,1]$ is the probability of the deterministic possibility $x$.
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@@ -29,7 +29,7 @@ A vector $d \in \mathbb{R}^n$ is a valid probability distribution iff $\sum d_i
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-This vector has $n$ entries, where each entry corresponds to a classical possibility $X$ and the probability of $X$ is $\Pr[X] = d_i$. The sum over all probabilities has to be $1$ and each entry needs to be nonnegative in order to be a valid probability.
+This vector has $n$ entries, where each entry corresponds to a deterministic possibility $X$ and the probability of $X$ is $\Pr[X] = d_i$. The sum over all probabilities has to be $1$ and each entry needs to be nonnegative in order to be a valid probability.
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@@ -40,7 +40,7 @@ For a coin flip the probability distribution would be $d_{\text{coin}} \in \math
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## Example (continued): Random 2-bit number
-Recall your random 2-bit number generator from above. Imagine your generator outputs each classical possibility with equal probability, except for the possibility $00$, which is never generated. The corresponding probability distribution would be
+Recall your random 2-bit number generator from above. Imagine your generator outputs each deterministic possibility with equal probability, except for the possibility $00$, which is never generated. The corresponding probability distribution would be
$$
d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix}
$$
@@ -49,7 +49,7 @@ $$
## Probabilistic processes
With a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a *probabilistic process*.
-A probabilistic process is a collection of $n$ probability distributions, where for each classical possibility $i$ there is a probability distribution $a_i$. This means, that if the system is in classical possibility $i$ before the process is applied, the system will afterwards be distributed according to $a_i$. We can write this as a matrix, where each column is a probability distribution $a_i$.
+A probabilistic process is a collection of $n$ probability distributions, where for each deterministic possibility $i$ there is a probability distribution $a_i$. This means, that if the system is in deterministic possibility $i$ before the process is applied, the system will afterwards be distributed according to $a_i$. We can write this as a matrix, where each column is a probability distribution $a_i$.
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::: {.definition #def-prob-process}
@@ -66,7 +66,7 @@ From @def-prob-distribution we know that a valid probability distribution $a$ ha
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## Example (continued): Random 2-bit number
-Imagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of $\frac{1}{3}$. The probability distributions for each of the classical possibility would then be
+Imagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of $\frac{1}{3}$. The probability distributions for each of the deterministic possibility would then be
$$
a_{00} = \begin{pmatrix} \frac{2}{3} \\ 0 \\ 0 \\ \frac{1}{3} \end{pmatrix}, a_{01} =\begin{pmatrix} 0 \\ \frac{2}{3} \\ \frac{1}{3} \\ 0 \end{pmatrix}, a_{10} =\begin{pmatrix} 0 \\ \frac{1}{3} \\ \frac{2}{3} \\ 0 \end{pmatrix}, a_{11} = \begin{pmatrix} \frac{1}{3} \\ 0 \\ 0 \\ \frac{2}{3} \end{pmatrix}
$$