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Quantum Cryptography
- 3. Public‐key cryptography relies on P≠NP (unproven) and can be broken by quantum computers. Ideal quantum cryptography is provably secure based on quantum mechanics. 1. Introduction - History: Stephen Wiesner ca. 1970; Charles Bennett, Gilles Brassard 1984 - neglecting engineering problems
- 6. Vernam cipher 0 0 1 1 0 1 1 0 1 0 1the plaintext: 0 1 1 1 0 0 1 0 0 1 1the pad: XOR = 0 1 0 0 0 1 0 0 1 1 0the ciphertext: - also "one-time-pad" - provably secure, i.e. ciphertext provides no information about the plaintext (no cryptanalysis possible) - problem: key distribution -> quantum cryptography generates a secret key
- 7. Vernam cipher 0 0 1 1 0 1 1 0 1 0 1the plaintext: 0 1 1 1 0 0 1 0 0 1 1the pad: XOR = 0 1 0 0 0 1 0 0 1 1 0the ciphertext: secure - also "one-time-pad" - provably secure, i.e. ciphertext provides no information about the plaintext (no cryptanalysis possible) - problem: key distribution -> quantum cryptography generates a secret key
- 8. Vernam cipher 0 0 1 1 0 1 1 0 1 0 1the plaintext: 0 1 1 1 0 0 1 0 0 1 1the pad: XOR = 0 1 0 0 0 1 0 0 1 1 0the ciphertext: secure truly random - also "one-time-pad" - provably secure, i.e. ciphertext provides no information about the plaintext (no cryptanalysis possible) - problem: key distribution -> quantum cryptography generates a secret key
- 9. Vernam cipher 0 0 1 1 0 1 1 0 1 0 1the plaintext: 0 1 1 1 0 0 1 0 0 1 1the pad: XOR = 0 1 0 0 0 1 0 0 1 1 0the ciphertext: used once secure truly random - also "one-time-pad" - provably secure, i.e. ciphertext provides no information about the plaintext (no cryptanalysis possible) - problem: key distribution -> quantum cryptography generates a secret key
- 12. 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 Alice's bases: Alice's states: Alice's key: ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ truly random BB84
- 13. 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 Alice's bases: Alice's states: Alice's key: ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ truly random truly random BB84
- 14. 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 Alice's bases: Alice's states: Alice's key: ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ truly random truly random truly random BB84
- 15. 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 Alice's bases: Alice's states: Alice's key: ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ truly random truly random truly random BB84
- 16. 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 Alice's bases: Alice's states: Alice's key: ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ truly random truly random truly random keep 50% BB84
- 17. 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 Alice's key: Eve's states: Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Alice's states: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ BB84 - Alice's and Eve's bases omitted - probability tree - also: transmission error
- 18. 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 Alice's key: Eve's states: Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Alice's states: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ BB84 - Alice's and Eve's bases omitted - probability tree - also: transmission error
- 19. 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 Alice's key: Eve's states: Bob's key: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ 25% error Alice's states: ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ Bob's bases: Bob's states: ↕ ↕ ↕ ↕ ↕ ✕✕ ✕ ✕ ✕ ✕ ✕ ✕ ↕ ↕ ↕ BB84 - Alice's and Eve's bases omitted - probability tree - also: transmission error
- 20. Attacks eavesdropping ➙ man‐in‐the‐middle ➙ multiple photons ➙ light pulse ➙ bad randomness ➙ introduces error use EPR work thoroughly work thoroughly work thoroughly - quantum phenomena - "strong" man-in-the-middle - EPR or: Lamport signatures (believed to be secure against quantum computers)
- 21. Attacks eavesdropping ➙ man‐in‐the‐middle ➙ multiple photons ➙ light pulse ➙ bad randomness ➙ introduces error use EPR work thoroughly work thoroughly work thoroughly no cloning - quantum phenomena - "strong" man-in-the-middle - EPR or: Lamport signatures (believed to be secure against quantum computers)
- 22. Attacks eavesdropping ➙ man‐in‐the‐middle ➙ multiple photons ➙ light pulse ➙ bad randomness ➙ introduces error use EPR work thoroughly work thoroughly work thoroughly use QM no cloning - quantum phenomena - "strong" man-in-the-middle - EPR or: Lamport signatures (believed to be secure against quantum computers)
- 24. Hilbert spaces Quantum states are described by unit vectors in a complex vector space with an inner product. a vector: an inner product: orthogonality: ψ|φ = 0 the norm: |ψ = ψ|ψ ψ|φ |ψ
- 25. Hilbert spaces {|0 , |1 } We consider the computational basis. a vector: an inner product: orthogonality: the norm: |ψ = α|0 + β|1 ψ|φ = α∗ α + β∗ β |ψ = |α|2 + |β|2 α∗ α + β∗ β = 0 -> blackboard
- 26. 0|1 = 1 0 0 1 = 0 0|+ = 1 0 1√ 2 1√ 2 = 1 √ 2 = 0 |0 = 1 · |0 + 0 · |1 |1 = 0 · |0 + 1 · |1 |+ = |0 + |1 √ 2 |− = |0 − |1 √ 2 |− = 1 √ 2 2 + − 1 √ 2 2 = 1 2 + 1 2 = 1 blackboard: examples
- 27. Hilbert spaces the tensor product: the natural inner product here: ψ ⊗ φ|ψ ⊗ φ = ψ|ψ φ|φ |ψ ⊗ |φ ≡ |ψφ -> blackboard
- 28. {|0 , |1 } {|00 , |01 , |10 , |11 } 11|01 = 1|0 1|1 = 0 · 1 = 0 blackboard: examples
- 29. an operator: its Hermitian adjoint: a unitary operator: Unitary operators U† U = I U† ψ|φ = ψ|Uφ Unitary operators describe the evolution of closed quantum systems. = αU|ψ + βU|φ U(α|ψ + β|φ ) -> blackboard
- 30. X|0 ≡ 0 1 1 0 1 0 = 0 1 ≡ |1 Y † Y ≡ 0 i∗ −i∗ 0 0 −i i 0 = 1 0 0 1 ≡ I blackboard: examples
- 32. The no‐cloning theorem It is impossible to copy an arbitrary unknown quantum state. U(|ψ ⊗ |s ) = |ψ ⊗ |ψ
- 33. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ - cloning possible if all states are equal or orthogonal
- 34. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ - cloning possible if all states are equal or orthogonal
- 35. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ - cloning possible if all states are equal or orthogonal
- 36. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ ⊗ s|φ ⊗ s = ψ ⊗ ψ|φ ⊗ φ - cloning possible if all states are equal or orthogonal
- 37. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ ⊗ s|φ ⊗ s = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ|φ s|s = ψ|φ ψ|φ - cloning possible if all states are equal or orthogonal
- 38. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ ⊗ s|φ ⊗ s = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ|φ s|s = ψ|φ ψ|φ ⇒ ψ|φ = ψ|φ 2 - cloning possible if all states are equal or orthogonal
- 39. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ ⊗ s|φ ⊗ s = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ|φ s|s = ψ|φ ψ|φ ⇒ ψ|φ = ψ|φ 2 ⇒ ψ|φ = 1 ∨ ψ|φ = 0 - cloning possible if all states are equal or orthogonal
- 40. The no‐cloning theorem U(|ψ ⊗ |s ) = |ψ ⊗ |ψ U(|φ ⊗ |s ) = |φ ⊗ |φ contra- diction ⇒ U(ψ ⊗ s)|U(φ ⊗ s) = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ ⊗ s|φ ⊗ s = ψ ⊗ ψ|φ ⊗ φ ⇒ ψ|φ s|s = ψ|φ ψ|φ ⇒ ψ|φ = ψ|φ 2 ⇒ ψ|φ = 1 ∨ ψ|φ = 0 - cloning possible if all states are equal or orthogonal
- 41. |ψ = α|0 + β|1 measured state with probability |α|2 |β|2 |0 |1 We cannot determine an arbitrary state. Measurement changes the state. Measurement - for unit vectors, probabilities sum to one
- 43. Measurement The completeness equation expresses that the probabilities sum to one. ψ| i M† i Mi|ψ = ψ|I|ψ = ψ|ψ = 1 ⇒ i p(i) = i ψ|M† i Mi|ψ = i M† i Mi = I
- 44. Indistinguishability of non‐orthogonal states There is no measurement to distinguish between any non‐ orthogonal quantum states. |ψ1 |ψ2 ψ1|ψ2 = 0 Ei := f(j)=i M† j Mj
- 45. Indistinguishability of non‐orthogonal states ψ1|E1|ψ1 = 1 ψ2|E2|ψ2 = 1
- 46. Indistinguishability of non‐orthogonal states ψ1|E1|ψ1 = 1 ψ2|E2|ψ2 = 1 i ψ1|Ei|ψ1 = ψ1|I|ψ1 = ψ1|ψ1 = 1
- 47. Indistinguishability of non‐orthogonal states ψ1|E1|ψ1 = 1 ψ2|E2|ψ2 = 1 ⇒ ψ1|E2|ψ1 = 0 ⇒ √ E2|ψ1 = 0 i ψ1|Ei|ψ1 = ψ1|I|ψ1 = ψ1|ψ1 = 1
- 48. Indistinguishability of non‐orthogonal states ψ1|E1|ψ1 = 1 ψ2|E2|ψ2 = 1 ⇒ ψ1|E2|ψ1 = 0 ⇒ √ E2|ψ1 = 0 |ψ2 = α|ψ1 + β|φ |β| < 1 i ψ1|Ei|ψ1 = ψ1|I|ψ1 = ψ1|ψ1 = 1
- 49. Indistinguishability of non‐orthogonal states ψ1|E1|ψ1 = 1 ψ2|E2|ψ2 = 1 ⇒ ψ1|E2|ψ1 = 0 ⇒ √ E2|ψ1 = 0 |ψ2 = α|ψ1 + β|φ |β| < 1 i ψ1|Ei|ψ1 = ψ1|I|ψ1 = ψ1|ψ1 = 1 ⇒ √ E2|ψ2 = α √ E2|ψ1 + β √ E2|φ
- 51. Indistinguishability of non‐orthogonal states ψ2|E2|ψ2 = α∗ α ψ1|E2|ψ1 + α∗ β ψ1|E2|φ +β∗ α φ|E2|ψ1 + β∗ β φ|E2|φ
- 52. Indistinguishability of non‐orthogonal states = |β|2 φ|E2|φ ψ2|E2|ψ2 = α∗ α ψ1|E2|ψ1 + α∗ β ψ1|E2|φ +β∗ α φ|E2|ψ1 + β∗ β φ|E2|φ
- 53. Indistinguishability of non‐orthogonal states = |β|2 φ|E2|φ ≤ |β|2 i φ|Ei|φ = |β|2 φ|I|φ ψ2|E2|ψ2 = α∗ α ψ1|E2|ψ1 + α∗ β ψ1|E2|φ +β∗ α φ|E2|ψ1 + β∗ β φ|E2|φ
- 54. Indistinguishability of non‐orthogonal states = |β|2 φ|E2|φ ≤ |β|2 i φ|Ei|φ = |β|2 φ|I|φ = |β|2 φ|φ = |β|2 < 1 ψ2|E2|ψ2 = α∗ α ψ1|E2|ψ1 + α∗ β ψ1|E2|φ +β∗ α φ|E2|ψ1 + β∗ β φ|E2|φ
- 55. Indistinguishability of non‐orthogonal states = |β|2 φ|E2|φ ≤ |β|2 i φ|Ei|φ = |β|2 φ|I|φ = |β|2 φ|φ = |β|2 < 1 contra- diction ψ2|E2|ψ2 = α∗ α ψ1|E2|ψ1 + α∗ β ψ1|E2|φ +β∗ α φ|E2|ψ1 + β∗ β φ|E2|φ
- 56. QC Revisited 1 0 ↕ ↕ ✕ ✕ ↕ ↕ states bases A state prepared in one basis and measured in a non‐orthogonal one is undetermined. - remember: photons polarized in different orientations - now: vectors in non-orthogonal bases
- 57. QC Revisited states bases 0 1 {|+ , |− }{|0 , |1 } |0 |1 |+ |− |+ := |0 + |1 √ 2 |− := |0 − |1 √ 2 - draw in 2D
- 59. QC Revisited |− = |0 − |1 √ 2 Measuring states in a non‐orthogonal basis:
- 60. QC Revisited |− = 1 √ 2 |0 + − 1 √ 2 |1 |− = |0 − |1 √ 2 Measuring states in a non‐orthogonal basis:
- 61. QC Revisited |− = 1 √ 2 |0 + − 1 √ 2 |1 |− = |0 − |1 √ 2 Measuring states in a non‐orthogonal basis: p(0|−) = 1 √ 2 2 = 1 2
- 62. QC Revisited |− = 1 √ 2 |0 + − 1 √ 2 |1 |− = |0 − |1 √ 2 Measuring states in a non‐orthogonal basis: p(0|−) = 1 √ 2 2 = 1 2 = − 1 √ 2 2 = p(1|−)
- 63. 4. Usage Using optical fiber (USA): 148.7 km Through the air (EU): 144 km SECOQC - state of the art - several companies, lots of research - big EU project