Random Numbers Certified by Bell's Theorem by Pironio and 10 other authors

from 6 different universities and institutions.

A brief introduction by David Kemp

Licensed under a Creative Commons Attribution 3.0 Licensehttp://creativecommons.org/licenses/by/3.0/

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.

 

Bell’s Inequality

• Published by John Stewart Bell in 1964• Proved “reality not separable from observation”.• Einstein died in 1955 believing reality must be

separable from observation (see EPR paradox).

Simple Inequality Example

Room full of people:•Only one right-handed male.•Only one blue-eyed female.•How many (max) blue-eyed right-handers?– At most one of them can be male.– At most one of them can be female.– Hence a maximum of 2.

Simple Inequality as Venn Diagram

Female

Right-handed

Blue-eyed

Size(blue-eyed right-handers) ≤ Size(blue-eyed females) + Size(right-handed males)

In Probabilistic Terms

Pr(blue-eyed right-hander) ≤Pr(blue-eyed female) + Pr(right-handed male)

What if your data is incomplete?

• Suppose you can only ask each person one of:– Are you a blue-eyed right-hander?– Are you a blue-eyed female?– Are you a right-handed male?

What if your data is incomplete?

•Suppose you can only ask each person one of:–Are you a blue-eyed right-hander?–Are you a blue-eyed female?–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!

Identical Twins

• Second room with a twin of everyone in first.• No communication between the rooms.• Ask every person only one of these questions:– Are you blue-eyed?– Are you left-handed?– Are you female?

• Sometimes ask both twins the same question.• Only way to avoid lie detection is prior agreement.

Bell’s Inequality

IF

•A, B and C are binary properties,•A', B' and C' are properties of twin,• Pr(x & not(x')) = 0.

THEN

Pr(A & not(B')) + Pr(B & not(C')) ≥ Pr(A & not(C'))

Pairs of Magic Boxes

A B C A B C

Three buttons and two lights on each box

Magic Box PairsA B C

Can only press one button on each box.

A B C

Magic Box Properties

Whenever same button is pressedon both boxes:

Same light flashes

A B C A B C

•Pr(A:red & B:yellow) ≈ 7.5%•Pr(B:red & C:yellow) ≈ 7.5 %•Pr(A:red & C:yellow) ≈ 25%

Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)

A B C A B CMagic Box Properties

But surely that is impossible!!!

Image by Wild Guru Larry. Some rights reserved.

http://www.flickr.com/photos/wentzelepsy/4435803492/

Relies on Quantum Entanglement

Entangled Electrons

• Measure Electron Spin Direction• Spin-up/down instead of clockwise/anti-clockwise• Result seems to be completely random.• Can measure from different angles.–Eg. Vertically, horizontally, and 45º to horizon.

• Measurement disturbs the electron.–Cannot reliably measure same electron from different angles.

Entangled Electrons

Let buttons A, B and C, measure at 0º, 45º and 90º•Spin-up = red•Spin-down = yellow

Entangled Electrons

Measure entangled electrons from same direction:

100% Correlation

(Press same button on both boxes and lights agree)

Entangled Electrons

• A:red & C:red 25%• A:yellow& C:yellow 25%• A:red & C:yellow 25%• A:yellow & C:red 25%

Measure entangled electrons at 90º to each other:50% Correlation

Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)

Entangled Electrons

• A:red & B:red ≈ 42.5%• A:yellow& B:yellow ≈ 42.5%• A:red & B:yellow ≈ 7.5%• A:yellow & B:red ≈ 7.5%

Measure entangled electrons at 45º to each other:≈ 85% Correlation

Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)

Entangled Electrons

• B:red & C:red ≈ 42.5%• B:yellow& C:yellow ≈ 42.5%• B:red & C:yellow ≈ 7.5%• B:yellow & C:red ≈ 7.5%

Measure entangled electrons at 45º to each other:≈ 85% Correlation

Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)

Quantum Entanglement is the only known process that

can violate Bell’s Inequality.

Verifiably Random

We can use these magic boxes to

generate verifiably random numbers

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!

Final Step

Result may not be uniformly random.

Use known randomness extraction techniques.

Practicalities

• Actually only need two buttons on each box (using the CHSH inequality).

• Difficult to prevent quantum entanglement quickly degenerating.

Caveats

• Need to prevent signaling between the boxes.• This “protocol is not yet proven to be universally-

composable against a full quantum adversary”.

References

•Pironio et al. Random Numbers Certied by Bell's Theorem, Nature v.464, p.1021 (2010).

•J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964)

•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