User:K10071995/sandbox

CNOT TEST ZUSTAND $$ $$
 * \psi \rangle = \frac{1}{\sqrt{2}} |00 \rangle + \frac{1}{\sqrt{2}} |01 \rangle

CONTROLLED NOT

$$CNOT: A \rightarrow A, B \rightarrow

\begin{cases} B&,  \lnot A \\ \lnot B&, A \end{cases}

$$

SIMPLE AND

$$ U_{\wedge}: A \mapsto A, B \mapsto A \wedge B, C \mapsto A \wedge B

$$

$$U_{\wedge}$$

Uand BEHAELT NORMIERUNG NICHT

$$ \begin{align} U_{\wedge}(| \psi \rangle) &= \frac{1}{\sqrt{2}} |11 \rangle + \frac{1}{\sqrt{2}} |11 \rangle = \frac{2}{\sqrt{2}} |11 \rangle \end{align}
 * A \rangle &= \frac{1}{\sqrt{2}} |10 \rangle + \frac{1}{\sqrt{2}} |11 \rangle \\

$$

$$ \Rightarrow p(|11|) = 2$$

$$ \begin{align} \end{align} $$
 * \psi \rangle = \frac{1}{\sqrt{2}} |0 \rangle + \frac{1}{\sqrt{2}} |1 \rangle \\
 * B \rangle = \frac{1}{\sqrt{2}} |0 \rangle - \frac{1}{\sqrt{2}} |1 \rangle

$$|A \rangle :$$ $$|B \rangle :$$

PROBABILITIES OF A and B

$$ |A \rangle : p(|0 \rangle) = p(|1 \rangle)  = |\frac{1}{\sqrt{2}}|^2 = \frac{1}{2} $$

$$ |B \rangle : p(|0 \rangle) = |\frac{1}{\sqrt{2}}|^2 = p(|1 \rangle)  =  |- \frac{1}{\sqrt{2}}|^2 = \frac{1}{2} $$

$$ \begin{align} \end{align} $$
 * A \rangle = \frac{1}{\sqrt{2}} |0 \rangle + \frac{1}{\sqrt{2}} |1 \rangle \\
 * B' \rangle = \frac{1}{\sqrt{2}} |0 \rangle - \frac{1}{\sqrt{2}} |1 \rangle

$$ \begin{align} p(|0 \rangle) & = |\alpha| ^2 \\ p(|1 \rangle) & = |\beta| ^2 \\ \end{align} $$

$$

\begin{align} p(|0 \rangle) & = |\alpha| ^2 \\ p(|1 \rangle) & = |\beta| ^2 \\ \end{align} $$

$$ 	\begin{align} \end{align} $$
 * \psi \rangle = \alpha * |0 \rangle + \beta * |1 \rangle

$$ \alpha, \beta \in \mathbb C, \qquad |\alpha|^2 + |\beta|^2 $$