collapse of superposition

main.tex

@@ -21,6 +21,7 @@
 \newtheorem{theorem}{Theorem}
 \newtheorem{lemma}{Lemma}
 \newtheorem{definition}{Definition}
+\newtheorem{remark}{Remark}
 
 \begin{document}
 
@@ -122,7 +123,8 @@ Extending this formalism to bit registers is actually fairly straight forward. S
   If $\mathbf{E} \coloneqq \parensc*{E_1, E_2, \dots, E_n}$ is the set of all possible outcomes of an experiment, then a superposition of probable outcomes is defined by:
   \begin{equation}
     E \coloneqq \sum_{i=1}^n p_i E_i \quad \text{with}\:\: p_i = P\parens*{E_i} \:\text{and}\:\: \sum_{i=1}^n p_i = 1
-  \end{equation}  
+  \end{equation}
+  The states (outcomes) in $\mathbf{E}$ are called basis states (outcomes). 
 \end{definition}
 
 As mentioned above, a superposition can not immediately be evaluated. It rather should be seen as a mathematical object holding incomplete knowledge about a certain property of some (stochastic) process, described by a random distribution $(p_i)_{i=1}^n$. Too actually evaluate a superposition, the missing information needs to be filled in by some kind of extra process e.g. performing an experiment, measuring an observable. After this extra information is filled in the property under consideration is fully known and the superposition \emph{collapses} to one of the actually realizable outcomes in $\mathbf{E}$. In this model a system can be in an uncertain state which only can be made concrete by some external influence like measuring an observable. This sounds quite abstract and especially the fact that a measurement could alter the state of a real physical system seems quite counterintuitive, but we will later see that this principle is actually grounded in reality.
@@ -130,9 +132,16 @@ As mentioned above, a superposition can not immediately be evaluated. It rather
 Let's consider the experiment of rolling a dice. Of course, for the observable \emph{number of eyes} the expected outcomes are $\mathbf{E} = \parensc{1, 2, \dots, 6}$. While the dice is still in the cup and in the state of being shaken number of eyes can not be reasonably determined, even if a transparent cup is being used. The dice is in a superposition $E = \sum_{i=1}^6 \frac{1}{6} \mathbf{i}$ of showing all numbers of eyes 1 to 6 with uniform probability $\frac{1}{6}$. In order to determine the number of eyes thrown, the dice needs to rest on a solid base, such that one side is evidently showing up. So by \emph{throwing the dice} we interfere with the system by stopping to shake the cup and placing the dice on a solid base (table). With the dice now laying on the table it is clearly showing only one number of eyes. The superposition collapsed!
 
 \begin{definition}{Collapse of Superposition}
-  
+  A state in superposition of basis states $\mathbf{E} = \parensc*{E_1, E_2, \dots, E_n}$ can be evaluated by collapsing it on one of its basis states. This is done by a measuring operator
+  \begin{equation}
+    M_{\mathbf{E}}\parens*{\sum_{i=1}^n p_i E_i} \coloneqq  E_i \quad\: \text{with probability}\:\: p_i 
+  \end{equation}
 \end{definition}
 
+\begin{remark}
+  The basis states are not unique. To see this, consider the experiment of rolling a dice. If the observable is \emph{the number of eyes} we have the basis states $\mathbf{E}_{\text{eye}} = \parensc*{\mathbf{i}}_{i=1}^6$. On the other hand, if the measurement is only supposed to distinguish between \emph{even or odd} numbers of eyes we have $\mathbf{E}_{\text{parity}} = \parensc*{\text{even}, \text{odd}}$. The corresponding measuring operators are $M_{\mathbf{E_{\text{eye}}}}$ and $M_{\mathbf{E_{\text{parity}}}}$.
+\end{remark}
+
 \subsection{Bit Registers in Superposition}