Random Numbers Certified By Bells Theorem
- 1. Random Numbers Certified by Bell's Theoremby Pironio and 10 other authorsfrom 6 different universities and institutions.A brief introduction by David KempLicensed under a Creative Commons Attribution 3.0 Licensehttp://creativecommons.org/licenses/by/3.0/
- 2. Why?Pseudo random numbers may have patterns known to attackers.There is need for a source of randomness that can be verified to be genuinely random.
- 3. Bell’s InequalityPublished by John Stewart Bell in 1964
- 4. Proved “reality not separable from observation”.
- 5. Einstein died in 1955 believing reality must be separable from observation (see EPR paradox).Simple Inequality ExampleRoom full of people:Only one right-handed male.
- 6. Only one blue-eyed female.
- 7. How many (max) blue-eyed right-handers?
- 8. At most one of them can be male.
- 9. At most one of them can be female.
- 10. Hence a maximum of 2.Right-handedBlue-eyedFemaleSimple Inequality as Venn DiagramSize(blue-eyed right-handers) ≤ Size(blue-eyed females) + Size(right-handed males)
- 11. In Probabilistic TermsPr(blue-eyed right-hander) ≤Pr(blue-eyed female) + Pr(right-handed male)
- 12. What if your data is incomplete?Suppose you can only ask each person one of:
- 13. Are you a blue-eyed right-hander?
- 14. Are you a blue-eyed female?
- 15. Are you a right-handed male?What if your data is incomplete?Suppose you can only ask each person one of:
- 16. Are you a blue-eyed right-hander?
- 17. Are you a blue-eyed female?
- 18. Are you a right-handed male?Pr(blue-eyed right-hander) ≤Pr(blue-eyed female) + Pr(right-handed male) ONLY IF THEY DO NOT LIE!
- 19. Identical TwinsSecond room with a twin of everyone in first.
- 20. No communication between the rooms.
- 21. Ask every person only one of these questions:
- 24. Are you female?
- 25. Sometimes ask both twins the same question.
- 26. Only way to avoid lie detection is prior agreement.Bell’s InequalityIFA, B and C are binary properties,
- 27. A', B' and C' are properties of twin,
- 28. Pr(x & not(x')) = 0.THENPr(A & not(B')) + Pr(B & not(C')) ≥ Pr(A & not(C'))
- 29. ABCABCPairs of Magic BoxesThree buttons and two lights on each box
- 30. ABCABCMagic Box PairsCan only press one button on each box.
- 31. ABCABCMagic Box PropertiesWhenever same button is pressedon both boxes:Same light flashes
- 32. ABCABCMagic Box PropertiesPr(A:red & B:yellow) ≈ 7.5%
- 33. Pr(B:red & C:yellow) ≈ 7.5 %
- 34. Pr(A:red & C:yellow) ≈ 25%Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
- 35. But surely that is impossible!!!Image by Wild Guru Larry. Some rights reserved.http://www.flickr.com/photos/wentzelepsy/4435803492/
- 36. Relies on Quantum Entanglement
- 37. Entangled ElectronsMeasure Electron Spin Direction
- 38. Spin-up/down instead of clockwise/anti-clockwise
- 39. Result seems to be completely random.
- 40. Can measure from different angles.
- 41. Eg. Vertically, horizontally, and 45º to horizon.
- 42. Measurement disturbs the electron.
- 43. Cannot reliably measure same electron from different angles.Entangled ElectronsLet buttons A, B and C, measure at 0º, 45º and 90ºSpin-up = redSpin-down = yellow
- 44. Entangled ElectronsMeasure entangled electrons from same direction:100% Correlation(Press same button on both boxes and lights agree)
- 45. Entangled ElectronsMeasure entangled electrons at 90º to each other:50% CorrelationA:red & C:red 25%
- 47. A:red & C:yellow 25%
- 48. A:yellow & C:red 25%Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
- 49. Entangled ElectronsMeasure entangled electrons at 45º to each other:≈ 85% CorrelationA:red & B:red ≈ 42.5%
- 50. A:yellow& B:yellow ≈ 42.5%
- 51. A:red & B:yellow ≈ 7.5%
- 52. A:yellow & B:red ≈ 7.5%Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
- 53. Entangled ElectronsMeasure entangled electrons at 45º to each other:≈ 85% CorrelationB:red & C:red ≈ 42.5%
- 54. B:yellow& C:yellow ≈ 42.5%
- 55. B:red & C:yellow ≈ 7.5%
- 56. B:yellow & C:red ≈ 7.5%Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
- 57. Quantum Entanglement is the only known process thatcan violate Bell’s Inequality.
- 58. Verifiably RandomWe can use these magic boxes togenerate verifiably random numbers
- 59. Selecting buttons A, B and C.How do you select buttons A, B and C?Need initial private random seed.Does not need to be strongly random!
- 60. Final StepResult may not be uniformly random.Use known randomness extraction techniques.
- 61. PracticalitiesActually only need two buttons on each box (using the CHSH inequality).
- 62. Difficult to prevent quantum entanglement quickly degenerating.CaveatsNeed to prevent signaling between the boxes.
- 63. This “protocol is not yet proven to be universally-composable against a full quantum adversary”.ReferencesPironio et al. Random Numbers Certied by Bell's Theorem, Nature v.464, p.1021 (2010).
- 64. J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964)
- 65. A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777–780 (1935)Schrödinger's Cat