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Andreas Tsouchlos 2025-12-04 19:41:45 +01:00
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src/fundamentals/main.tex
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@ -84,7 +84,7 @@ as a vector}
The so called Bra-ket or Dirac notation is especially appropriate, The so called Bra-ket or Dirac notation is especially appropriate,
having been proposed by Paul Dirac in 1939 for the express purpose having been proposed by Paul Dirac in 1939 for the express purpose
of simplifying quantum mechanical notation \cite{dirac_new_1939}. of simplifying quantum mechanical notation \cite{dirac_new_1939}.
Two new symbols are introduced, \emph{bra}s $\bra{\cdot}$ and Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and
\emph{ket}s $\ket{\cdot}$. \emph{ket}s $\ket{\cdot}$.
Kets denote ordinary vectors, while bras denote their Hermitian conjugates. Kets denote ordinary vectors, while bras denote their Hermitian conjugates.
For example, two vectors specified by the labels $a$ and $b$ For example, two vectors specified by the labels $a$ and $b$
@ -119,7 +119,8 @@ Their inner product is $\braket{a\vert b}$.
.% .%
\end{align*}% \end{align*}%
% %
\ldots When not ambiguous in the context, the tensor product symbol may be omitted, e.g. \ldots When not ambiguous in the context, the tensor product
symbol may be omitted, e.g.,
\begin{align*} \begin{align*}
\ket{0} \otimes \ket{0} = \ket{0}\ket{0} \ket{0} \otimes \ket{0} = \ket{0}\ket{0}
.% .%
@ -128,7 +129,7 @@ Their inner product is $\braket{a\vert b}$.
As we will see, the core concept that gives quantum computing its As we will see, the core concept that gives quantum computing its
power is entanglement. When two quantum mechanical systems are power is entanglement. When two quantum mechanical systems are
entangled, measuring the state of one will collapse the state of the other. entangled, measuring the state of one will collapse that of the other.
Take for example two subsystems with the overall state Take for example two subsystems with the overall state
% %
\begin{align*} \begin{align*}
@ -137,9 +138,9 @@ Take for example two subsystems with the overall state
.% .%
\end{align*} \end{align*}
% %
If we measure the state of the first subsystem as $\ket{0}$, we can If we measure the first subsystem as being in $\ket{0}$, we can
be certain that a measurement of the second subsystem will also yield $\ket{0}$. be certain that a measurement of the second subsystem will also yield $\ket{0}$.
For brevity, we write the state of the combined system as% Introducing a new notation for entangled states, we can write%
% %
\begin{align*} \begin{align*}
\ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right) \ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)