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
-
\(X(0)= I\)
-
\(X(\frac{\pi }{2}) =|+\rangle \langle +|+e^{\frac{\pi }{2}i} |-\rangle \langle -| = |+\rangle \langle +|- i|-\rangle \langle -| \)$$\begin{aligned}= & {} \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right] - i \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,-\frac{1}{\sqrt{2}} \right] \\= & {} \left[ \begin{array}{ll} \frac{1}{2} &{}\quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] - i\left[ \begin{array}{ll} \frac{1}{2} &{} \quad -\frac{1}{2} \\ -\frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] = \frac{1}{2} \left[ \begin{array}{ll} 1-i &{}\quad 1+i \\ 1+i &{}\quad 1-i \end{array} \right] . \end{aligned}$$
-
\(X(\pi )= X\)
-
\(X(\frac{3\pi }{2}) =|+\rangle \langle +|+e^{\frac{3\pi }{2}i} |-\rangle \langle -| = |+\rangle \langle +|+ i|-\rangle \langle -| \)$$\begin{aligned}= & {} \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}} \right] + i \left[ \begin{array}{l} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{array} \right] \left[ \frac{1}{\sqrt{2}} ,-\frac{1}{\sqrt{2}} \right] \\= & {} \left[ \begin{array}{ll} \frac{1}{2} &{} \quad \frac{1}{2} \\ \frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] + i\left[ \begin{array}{ll} \frac{1}{2} &{} \quad -\frac{1}{2} \\ -\frac{1}{2} &{}\quad \frac{1}{2} \end{array} \right] \\= & {} \frac{1}{2} \left[ \begin{array}{ll} 1+i &{} \quad 1-i \\ 1-i &{} \quad 1+i \end{array} \right] . \end{aligned}$$
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) |
\(X(\frac{\pi }{2})\)
|
\(X(\pi )\)
|
\(X(\frac{3\pi }{2})\)
| |
\(|0\rangle \)
|
\(|0\rangle \)
|
\(|-i\rangle \)
|
\(|1\rangle \)
|
\(|+i\rangle \)
|
\(|1\rangle \)
|
\(|1\rangle \)
|
\(|+i\rangle \)
|
\(|0\rangle \)
|
\(|-i\rangle \)
|
\(|+i\rangle \)
|
\(|+i\rangle \)
|
\(|0\rangle \)
|
\(|-i\rangle \)
|
\(|1\rangle \)
|
\(|-i\rangle \)
|
\(|-i\rangle \)
|
\(|1\rangle \)
|
\(|+i\rangle \)
|
\(|0\rangle \)
|