![Schrödinger 1932
The claim ... has the strange consequence that the
Ψ-function of a system [System I] is changed by the
performance of a measurement on a different, far
separated system [System II] ...
Schrödinger Archiv,Wien](https://image.slidesharecdn.com/encryptingwithentanglementmatthiaschristandl-130702085959-phpapp01/85/Encrypting-with-entanglement-matthias-christandl-5-320.jpg)
![Schrödinger 1932 Schrödinger Archiv,Wien
This makes it a bit difficult to view the change in the
Ψ-function as a Naturvorgang.x)
...
x) the matter becomes even more strange, if we
perform a different measurement on [System II] ...](https://image.slidesharecdn.com/encryptingwithentanglementmatthiaschristandl-130702085959-phpapp01/85/Encrypting-with-entanglement-matthias-christandl-6-320.jpg)
![Algorithm: Extendible to k Bobs?
Yes almost not entangled
No entangled
Result: Algorithm is fast
2 Bobs
not entangled
Bobs
3 Bobs
k Bobs
11
|AB|
min(|A|,|B|
i,j |iijj|
|AB|
|00
O
log |A|
2
∞−
entanglement is the quantum analogue of the intrinsic information, which is defined
I(X;Y ↓Z) := inf
P ¯Z|Z
I(X; Y | ¯Z),
f random variables X, Y, Z [16]. The minimisation extends over all conditional prob-
butions mapping Z to ¯Z. It has been shown that the minimisation can be restricted
ariables ¯Z with size | ¯Z| = |Z|[17]. This implies that the minimum is achieved and in
all quantum states
√q/k
}
in practice
(semidefinite
programming)
in theory
(quasipolynomial-time)
Fernando Brandão,
Matthias Christandl
und JonYard
(2010)](https://image.slidesharecdn.com/encryptingwithentanglementmatthiaschristandl-130702085959-phpapp01/85/Encrypting-with-entanglement-matthias-christandl-53-320.jpg)
Encrypting with entanglement matthias christandl
- 1. Matthias Christandl Quantum Information Theory Institute for Theoretical Physics ETH Zurich Encrypting with Entanglement
- 2. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
- 3. Quantum Mechanics • Theory of the smallest particles • Big implications Stability of matter Fission and fusion of nuclei Hawking radiation ω✿✿✿✿✿✿ Photon
- 4. Entanglement - a Quantum Mechanical Phenomenon • Quantum mechanical correlations among two or more particles • „spooky action at a distance“ • „entanglement is not one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought“ Albert Einstein Erwin Schrödinger
- 5. Schrödinger 1932 The claim ... has the strange consequence that the Ψ-function of a system [System I] is changed by the performance of a measurement on a different, far separated system [System II] ... Schrödinger Archiv,Wien
- 6. Schrödinger 1932 Schrödinger Archiv,Wien This makes it a bit difficult to view the change in the Ψ-function as a Naturvorgang.x) ... x) the matter becomes even more strange, if we perform a different measurement on [System II] ...
- 7. Alice and Bob • Long distances • Communication of measurement results • Particles in the hands of • Each equipped with a laboratory Alice Bob and
- 8. Information • The bit = unit of information on/off heads/tails north pole/ south pole
- 9. Information • random bit child plays with switch toss of a coin travel lottery
- 10. Correlated Bits • Alice gets result Bob the opposite • Alice heads Bob tails Alice tails Bob heads • Random, but correlated bits
- 11. Qubit, the Quantum Bit • Unit of quantum information • Many possible states (dots on a sphere) • Example: photon polarisation . . ! # !# !$#%%!'# !(#%%!)# !*# !+# !,#
- 12. Qubit, the Quantum Bit • Unit of quantum information • Many possible states (dots on a sphere) • Example: photon polarisation . . ! # !# !$#%%!'# !(#%%!)# !*# !+# !,#
- 13. Qubit • We cannot measure accurately the state of the qubit. • We can only measure, if the state is in one of two antipodal points: • North or south pole? • Madrid or Wellington? • Bangkok or Lima?
- 14. Qubit • State: North pole Measurement: North or south pole? Result: North pole • State: Copenhagen Measurement: North or south pole? Result: Nordpol (Cos2 35°/2≈91%) • State: Singapur Measurement: North or south pole? Result: North pole (Cos2 90°/2=50%) •Measurement changes the state 50% 50%
- 18. Measurement: North or south pole Source Entangled Qubits 50%
- 19. Measurement: North or south pole Source Entangled Qubits 50% 50%
- 20. Measurement: North or south pole Source Entangled Qubits 50% 50% Bob‘s state=antipodal point just like a coin
- 24. Entangled Qubits 50% 50% Bob‘s state=antipodal point for every measurement „spooky action at a distance“ Measurement: Madrid or Wellington Source
- 25. Übersicht • Entanglement • Determinism? • Quantum Cryptography • Test for Entanglement message + key ---------------- = cipher
- 26. Determinism? • In classical physics, the measurement result exists before the performance of the measurement (realism) • Is there an element of reality which determines the measurement result in quantum mechanics? • No: Measurement results are inherently probabilistic. The world is not deterministic ? „God does not place dice“ Einstein, Podolsky and Rosen (1935) Bell (1967)
- 27. Bell‘s Inequality • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% Source a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1 a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1
- 28. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% Bell‘s Inequality a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1 100% 100% 100% 0% a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1
- 29. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1 Bell‘s Inequality Source a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1
- 30. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1 Bell‘s Inequality Source a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1 Cos2 45°/2 ≈ 85% Cos2 45°/2 ≈ 85% Cos2 45°/2 ≈ 85% 1-Cos2 135°/2≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1
- 31. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1 Bell‘s Inequality Source a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1
- 32. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1 Bell‘s Inequality Source a=0 or 1 y=0 or 1 b=0 or 1 x=0 or 1x=0 oder 1 must be confirmed in the experiment each measurement must yield a result (successful detection) Choice of x must be independent of y (locality) a≠b für x=0, y=0 a≠b für x=0, y=1 a≠b für x=1, y=0 a=b für x=1, y=1
- 33. Experimente • Photons,Aspect et al. (1982) Locality ✗ Detection ✗ • Photons, Gisin et al., Zeilinger et al.(1998) Locality ✓ Detection ✗ • Superconducting Qubits Wineland et al. (2001) Locality ✗ Detection ✓ • Locality and detection in one experiment?
- 34. Experimente • Photons,Aspect et al. (1982) Locality ✗ Detection ✗ • Photons, Gisin et al., Zeilinger et al.(1998) Locality ✓ Detection ✗ • Superconducting Qubits Wineland et al. (2001) Locality ✗ Detection ✓ • Locality and detection in one experiment? Indeterminism of the world! Security of quantum cryptography!
- 35. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
- 36. Quantum Cryptography • Measurement result is random and correlated • In principle, the measurement result has not existed before the measurement • Only Alice and Bob know the result Alice and Bob have a secret bit ? many secret bitsrepetition
- 37. Quantum Cryptography • Measurement result is random and correlated • In principle, the measurement result has not existed before the measurement • Only Alice and Bob know the result Alice and Bob have a secret bit ? keyrepetition
- 38. Quantum Cryptography • Measurement result is random and correlated • In principle, the measurement result has not existed before the measurement • Only Alice and Bob know the result Alice and Bob have a secret bit • Encrypting with Entanglement ? Ekert (1991) keyrepetition
- 39. Perfectly Secret Communication • Vernam (1926) Shannon (1949) message + key ---------------- = cipher cipher - key ---------------- = message
- 40. 00101 10100 01000 + 10011 01010 11010 ----------------------------- = 10010 11110 10010 10010 11110 10010 - 10011 01010 11010 ----------------------------- =00101 10100 01000 Perfectly Secret Communication • Vernam (1926) Shannon (1949) • Perfect secrecy • Commercial: idQuantique, MagiQ Technologies
- 41. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
- 42. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key? 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 43. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key? 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 44. Encrypting with Entanglement • Theory • Experiment - Noise 0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 45. Encrypting with Entanglement • Theory • Experiment - Noise 0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 46. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key? 0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 47. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key? Test for Entanglement 0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06 0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0
- 48. Monogamy of Entanglement Alice strongly entangled with Bob 1 ➭ Alice little entangled with Bob 2 Bob 1 Bob 2 Alice
- 49. Monogamy of Entanglement Alice strongly entangled with Bob 1 ➭ Alice little entangled with Bob 2 . . . ➭ Alice little entangled with Bob k Bob 1 Bob 2 Bob k Alice
- 50. Monogamy of Entanglement Bob 1 Bob 2 Bob k Alice
- 51. Monogamy of Entanglement Bob 1 Bob 2 Bob k Alice Given: State of Alice and Bob 1 Question: Can Alice be entangled with k Bobs in equal fashion? Answer: Yes: State is almost not entangled (almost not = ) No: State is entangled 1 √ k
- 52. Mathematical Formulation extendible to k Bobs Frobenius (Euclidian) norm n = |A|2 |B| O log |A| 2 = eO(−2 log |A| log |B|) eO(−2 log |A| log |B|) eO(|A|2|B|2 log −1) eO(|A|2|B|2) eO(log |A| log |B|) ||X|| := √ tr X†X ||X||1 := tr √ X†X 1 0 0 0 0 0 0 0 number of Alice‘s qubits min σAB ||ρAB − σAB|| ≤ c q k Fernando Brandão, Matthias Christandl und JonYard (2010) not entangled
- 53. Algorithm: Extendible to k Bobs? Yes almost not entangled No entangled Result: Algorithm is fast 2 Bobs not entangled Bobs 3 Bobs k Bobs 11 |AB| min(|A|,|B| i,j |iijj| |AB| |00 O log |A| 2 ∞− entanglement is the quantum analogue of the intrinsic information, which is defined I(X;Y ↓Z) := inf P ¯Z|Z I(X; Y | ¯Z), f random variables X, Y, Z [16]. The minimisation extends over all conditional prob- butions mapping Z to ¯Z. It has been shown that the minimisation can be restricted ariables ¯Z with size | ¯Z| = |Z|[17]. This implies that the minimum is achieved and in all quantum states √q/k } in practice (semidefinite programming) in theory (quasipolynomial-time) Fernando Brandão, Matthias Christandl und JonYard (2010)
- 54. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
- 55. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
- 56. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle
- 57. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle
- 58. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
- 59. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
- 60. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
- 61. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
- 62. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality? Mathematical Tools Statistics of the Quanta Symmetries of the Quanta