# Quantum key distribution
- **Goal:**
- Creation of a private key over a public channel (not sending messages)
- Properties:
- needed: a quantum channel for transmitting qubits with low error rate
- detecting eavesdropping
- security is guaranteed by laws of quantum physics
- **Fundamental priciple:** No information without disturbing
- No-cloning Theorem
- Disturbance
- Attempting to distinguish between **Non-orthogonal** states
gaining information $\Rightarrow$ disturbance
- proof:
- assume $|\psi\rangle$ and $|\phi\rangle$'s information can be obtained by an unitary $U$ and an ancillary state $|u\rangle$
- $$|\psi\rangle|u\rangle\xrightarrow[]{U}|\psi\rangle|v\rangle \\
|\phi\rangle|u\rangle\xrightarrow[]{U}|\phi\rangle|v'\rangle$$
-
$$
\langle\psi|\phi\rangle\langle u|u\rangle=\langle\psi|\phi\rangle\langle v|v'\rangle
$$
since $|\psi\rangle$ and $|\phi\rangle$ are **non-orthogonal**, $\langle\psi|\phi\rangle\neq0$
- $\therefore\langle v|v'\rangle$
$v, v'$ are identical
- *No information about $|\psi\rangle$ and $|\phi\rangle$ is obtained.*

### BB84: transmission of single qubit
1. A chooses $(4+\delta)n$ bits data $a$
2. A chooses $(4+\delta)n$ bits basis $b$
A encodes data as

|a\b | 0| 1|
|--|--|--|
| 0|\|$0\rangle$ |\|$+\rangle$ |
| 1|\|$1\rangle$ |\|$-\rangle$ |
3. A sends the resulting state to B
4. B measures in random basis $b'$
- If use $X$-basis measures $\{|0\rangle,|1\rangle\}$, the outcome will be random $0,1$
5. A announce $b$
6. A and B compare $b$ and $b'$, discard $b_i\neq b'_i$
with high probability there are $2n$ bits left
7. A select $n$ check bits to detect eavesdropping, and tell B
8. A and B announce check bits
> no entanglement needed

### BB92: entanglement-based quantum cryptography
- A measures entangled state $1/\sqrt{2}(|00\rangle+|11\rangle)$
outcome $\{0,1\}$ will be random but perfectly correlated with B's measurement.
A measures $1\Rightarrow$ B measures $1$ if in the same basis
- A and B measure in different basis $\Rightarrow$ uncorrelated

1. prepare a few Bell pairs to perform **Bell test**
2. A and B measure their qubit in independently and randomly choosen bases.
3. discard the bits measured in different basis (same basis $\Rightarrow$ same measurement outcome)