Write first draft of core concepts and observables subsections
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@@ -72,3 +72,8 @@
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short=AWGN,
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long=additive white Gaussian noise
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}
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\DeclareAcronym{pdf}{
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short=PDF,
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long=probability density function
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}
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@@ -592,15 +592,30 @@ decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
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\section{Quantum Mechanics and Quantum Information Science}
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\label{sec:Quantum Mechanics and Quantum Information Science}
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% TODO: Should the brief intro to QC be made later on or here?
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Designing codes and decoders for \ac{qec} is generally performed on a
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layer of abstraction far removed from the quantum mechanical
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processes underlying the actual qubits.
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Nevertheless, having a fundamental understanding of the related
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quantum mechanical concepts is useful to understand the unique constraints
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of this field.
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The purpose of this section is to convey these concepts to the reader.
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%%%%%%%%%%%%%%%%
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\subsection{Core Concepts and Notation}
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\label{subsec:Notation}
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\ldots can be very elegantly expressed using the language of
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linear algebra.
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\todo{Mention that we model the state of a quantum mechanical system
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as a vector}
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% Wave functions
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In quantum mechanics, the evolution of a state of a particle over tme
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and space is described by a \emph{wave function} $\psi(x,t)$.
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The connection between this function and the world that we can observe
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is the fact that $\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of
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finding a praticle in that particular state.
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% Dirac notation
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A lot of the related mathematics can be very elegantly expressed
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using the language of linear algebra.
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The so called Bra-ket or Dirac notation is especially appropriate,
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having been proposed by Paul Dirac in 1939 for the express purpose
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of simplifying quantum mechanical notation \cite{dirac_new_1939}.
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@@ -611,9 +626,170 @@ For example, two vectors specified by the labels $a$ and $b$
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respectively are written as $\ket{a}$ and $\ket{b}$.
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Their inner product is $\braket{a\vert b}$.
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% Expressing wave functions using linear algebra
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We can model a wave function $\psi(x,t)$ as a linear combination of different
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\emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as%
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%
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\begin{align*}
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\psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t)
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.%
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\end{align*}
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To express this relation using linear algebra, we represent
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$\psi(x,t)$ and $e_n(x,t)$ as vectors $\ket{\psi}$ and $\ket{e_n}$.
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We write%
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%
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\begin{align*}
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\ket{\psi} = \sum_{n=1}^{\infty} c_n \ket{e_n}
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.%
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\end{align*}
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% Operators
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Another important notion is that of an \emph{operator}, a component
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that takes a function as an input and returns another function as an output.
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Operators are useful to describe the relations between different
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quantities relating to a particle.
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An example of this is the differential operator $\partial x$.
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%%%%%%%%%%%%%%%%
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\subsection{Observables}
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\label{subsec:Observables}
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% Observable quantities
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An \emph{observable quantity} $Q$ is \ldots .
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Due to the probabilistic nature of quantum mechanics, the result of a
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measurement is not deterministic.
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Thus, it is useful to consider the \emph{expected value} $\braket{Q}$
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of an observable quantity in addition to individual measurement results.
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% General expression for expected value of observable quantity
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If we know the wave function of a particle, we should be able to
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compute $\braket{Q}$ for any observable quantity we wish.
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It can be shown that for any $Q$, we can compute a
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corresponding operator $\hat{Q}$ such that%
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%
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\begin{align}
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\label{eq:gen_expr_Q_exp}
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\braket{Q} = \int_{-\infty}^{\infty} \psi^*(x,t) \hat{Q} \psi(x,t) dx
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.%
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\end{align}%
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%
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While the derivation of this relationship is out of the scope of this
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work, we can at least look at an example to illustrate it.
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Considering the position $Q = x$ of a particle and setting the observable
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operator to $\hat{Q} = x$, we can write%
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%
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\begin{align*}
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\braket{x} = \int_{-\infty}^{\infty} \psi^*(x,t) \cdot x \cdot \psi(x,t) dx
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= \int_{-\infty}^{\infty} x \lvert \psi(x,t) \rvert ^2 dx
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.%
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\end{align*}
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%
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Note that $\lvert \psi(x,t) \rvert $ represents the \ac{pdf} of
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finding a particle in a specific state. We immediately see that the
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formula simplifies to the direct calculation of the expected value.
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% Determinate states and eigenvalues
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% TODO: Introduce determinate states above
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% TODO: Nicer phrasing
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% TODO: Use different symbol for determinate states (not psi)
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% TODO: Fix equation
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Let us now examine how the observable operator $\hat{Q}$ relates to
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the determinate states that make up the overall superposition state
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of the particle.
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We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as%
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%
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\begin{align}
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\label{eq:gen_expr_Q_exp_lin}
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\braket{Q} = \braket{\psi \vert \hat{Q}\psi}
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.%
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\end{align}
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%
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\autoref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
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relationhip.
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The determinate states are inherently deterministic.
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To relate the two, we look at those states $\ket{\psi}$, where the
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variance of the measurements of $Q$ is zero. These are exactly the
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determinate states.%
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%
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\begin{align*}
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0 &\overset{!}{=} \braket{(Q - \braket{Q})^2}
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= \braket{\psi \vert (\hat{Q} - \braket{Q})^2 \psi} \\
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&= \braket{(Q - \braket{Q})\psi \vert (\hat{Q} - \braket{Q}) \psi} \\
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&= \lVert (Q - \braket{Q}) \psi \rVert^2 \\[3mm]
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&\hspace{-8mm}\Leftrightarrow (\hat{Q} - \braket{Q}) \ket{\psi} = 0 \\
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\label{eq:observable_eigenrelation}
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&\hspace{-8mm}\Leftrightarrow \hat{Q}\ket{\psi} = \braket{Q} \ket{\psi}
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.%
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\end{align*}%
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%
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Because we have assumed the variance to be zero, $\braket{Q}$ is now
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the deterministic measurement value corresponding to the determinate
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state $\ket{\psi}$.
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We can see that the determinate states are the \emph{eigenstates} of
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the observable operator $\hat{Q}$ and that the corresponding
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(deterministic) measurement values are the corresponding \emph{eigenvalues}.
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% Recap
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To summarize, we can mathematically express any observable quantity
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$Q$ using a corresponding operator $\hat{Q}$.
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This operator allows us to both compute the expected value of the
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observable using \autoref{eq:gen_expr_Q_exp_lin}, and describe the
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individual determinate states and corresponding measurement values
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using \autoref{eq:observable_eigenrelation}.
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%%%%%%%%%%%%%%%%
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\subsection{Projective Measurements}
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\label{subsec:Projective Measurements}
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%%%%%%%%%%%%%%%%
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\subsection{Qubits and Multi-Qubit States}
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\label{subsec:Qubits and Multi-Qubit States}
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\red{
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\begin{itemize}
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\item Projective measurements
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\begin{itemize}
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\item Using the determinate states as a basis for the
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superposition state
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\item The effect of using the obserrvable operator on
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the superposition state
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\item Modelling the observable operator as a series
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of projections
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\item The projection operator and its eigenvalues
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\end{itemize}
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\item Qubits and multi-qubit states
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\begin{itemize}
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\item The qubit
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\begin{itemize}
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\item Similar structure to classical
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computing: bits are modified with gates
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-> quantum bits are modified with quantum gates
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\end{itemize}
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\item The tensor product
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\item Information is not stored in the individual bit
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states but in the correlations / entanglement between them
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\item -> The size of the vector space
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\item The X,Z and Y operators
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\item (?) Notation of operators on multi-qubit states
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\end{itemize}
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\end{itemize}
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}
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\red{
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\begin{itemize}
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\item Representing wave functions as vectors (psi as label,
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building a vector space using basis functions)
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\end{itemize}
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}
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\red{\textbf{Tensor product}}
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\red{\ldots
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\todo{Introduce determinate state or use a different word?}
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Take for example two systems with the determinate states $\ket{0}$
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and $\ket{1}$. In general, the state of each can be written as the
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superposition%
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@@ -668,11 +844,6 @@ Introducing a new notation for entangled states, we can write%
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\end{align*}
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%
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\subsection{Projective Measurements}
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\label{subsec:Projective Measurements}
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% TODO: Write
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%%%%%%%%%%%%%%%%
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\subsection{Quantum Gates}
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\label{subsec:Quantum Gates}
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