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)