06.07.2017  Methodologies and Application  Ausgabe 19/2018 Open Access
Categorical quantum cryptography for access control in cloud computing
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 Soft Computing > Ausgabe 19/2018
1 Introduction
2 Quantum theory
2.1 Qubits
State  Operation  

I

H

X
 
\(0\rangle \)

\(0\rangle \)

\(+\rangle \)

\(1\rangle \)

\(1\rangle \)

\(1\rangle \)

\(\rangle \)

\(0\rangle \)

\(+\rangle \)

\(+\rangle \)

\(0\rangle \)

\(+\rangle \)

\(\rangle \)

\(\rangle \)

\(1\rangle \)

\(\rangle \)

2.2 Quantum gates
2.3 Measurement
2.4 Graphical calculus for quantum computation

\(X(0)= I\)

\(X(\frac{\pi }{2}) =+\rangle \langle ++e^{\frac{\pi }{2}i} \rangle \langle  = +\rangle \langle + i\rangle \langle  \)$$\begin{aligned}= & {} \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right]  i \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right] \\= & {} \left[ \begin{array}{ll} \frac{1}{2} &{}\quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right]  i\left[ \begin{array}{ll} \frac{1}{2} &{} \quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] = \frac{1}{2} \left[ \begin{array}{ll} 1i &{}\quad 1+i \\ 1+i &{}\quad 1i \end{array} \right] . \end{aligned}$$

\(X(\pi )= X\)

\(X(\frac{3\pi }{2}) =+\rangle \langle ++e^{\frac{3\pi }{2}i} \rangle \langle  = +\rangle \langle ++ i\rangle \langle  \)$$\begin{aligned}= & {} \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right] + i \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right] \\= & {} \left[ \begin{array}{ll} \frac{1}{2} &{} \quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] + i\left[ \begin{array}{ll} \frac{1}{2} &{} \quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] \\= & {} \frac{1}{2} \left[ \begin{array}{ll} 1+i &{} \quad 1i \\ 1i &{} \quad 1+i \end{array} \right] . \end{aligned}$$
3 Quantum cryptography for access control
3.1 Quantum key distribution

Alice lives in a country where the police open all mails.

Bob wants to send an object to Alice.

Bob has a strongbox which is big enough for several locks, but Alice does not have any key for any of those locks.
State  Operation  

X(0) 
\(X(\frac{\pi }{2})\)

\(X(\pi )\)

\(X(\frac{3\pi }{2})\)
 
\(0\rangle \)

\(0\rangle \)

\(i\rangle \)

\(1\rangle \)

\(+i\rangle \)

\(1\rangle \)

\(1\rangle \)

\(+i\rangle \)

\(0\rangle \)

\(i\rangle \)

\(+i\rangle \)

\(+i\rangle \)

\(0\rangle \)

\(i\rangle \)

\(1\rangle \)

\(i\rangle \)

\(i\rangle \)

\(1\rangle \)

\(+i\rangle \)

\(0\rangle \)
