[Quantum Information] Quantum key distribution

  Quantum Key Distribution

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Müller, Markus

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)