Matt Coudron and Henry Yuen 6845 121212 God does not play dice Albert Einstein Einstein stop telling God what to do Niels Bohr The Motivating Question Is it possible to test randomness ID: 606109
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Slide1
Some Limits on Non-Local Randomness Expansion
Matt Coudron and Henry Yuen6.84512/12/12
God does not play dice. --Albert Einstein
Einstein, stop telling God what to do.
--
Niels
BohrSlide2
The Motivating Question
Is it possible to test randomness?Slide3
The Motivating Question
Is it possible to test randomness?
100010100111100…..Slide4
The Motivating Question
Is it possible to test randomness?
111111111111111…..Slide5
Non-local games offers a way…
x
ϵ
{0,1}
y
ϵ
{0,1}
a
ϵ
{0,1}
b
ϵ
{0,1}
CHSH game
:
a+b
= x
Λ
y
Classical win probability: 75%
Quantum win probability:
~85%Slide6
Non-locality offers a way…
x
ϵ
{0,1}
y
ϵ
{0,1}
a
ϵ
{0,1}
b
ϵ
{0,1}
CHSH game
:
a+b
= x
Λ
y
Classical win probability: 75%
Quantum win probability:
~85%
Key insight:
if
the devices win the CHSH game
with > 75% success probability,
then
their outputs
must be randomized!Slide7
Non-locality offers a way…
[
Colbeck
‘10][PAM+ ‘10][VV
’11] devised
protocols that not only
certify
randomness, but also
expand
it!
1000101001
short random seed
0
1
1
0
1
0
0
1
1
0
1
0
0
11
01001
Referee tests outputs, and if test passes,
outputs are random!
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11111010101….
01000010100….
long pseudorandom input sequence
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
0
1
Referee feeds devices inputs and collects outputs in a streaming fashion.Slide8
Exponential certifiable randomness
Vazirani-Vidick
Protocol achieves
exponential
certifiable randomness expansion!
1000101001
n-
bit seed
0
0
0
0
0
0
Referee tests that the
devices win the CHSH
game ~85% of time
per block.
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
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0
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11111010101….
01000010100….
2
O
(
n
)
rounds
Bell block:
inputs are randomized
If the outputs pass the test, then they’re certified to have
2
O
(
n
)
bits of entropy!
Regular block:
inputs are
deterministic
Regular block:
inputs are
deterministicSlide9
And the obvious question is...
Can we do better?Doubly exponential?…
infinite expansion?Slide10
Our results
Upper boundsNonadaptive protocols performing “AND” tests, with perfect games: doubly exponential upper bound.
Nonadaptive (no signalling) protocols performing CHSH tests: exponential upper boundShows VV-like protocols and analysis are essentially optimal!
Lower bounds
A simplified VV protocol that achieves better
randomness rate
.Slide11
Definitions
Non-Adaptive“AND” TestPerfect GamesCHSH Tests
1000101001
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
0
1
Test
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11111010101….
01000010100….
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
0
1Slide12
Doubly Exponential Bound
Must exhibit a “cheating strategy” for Alice and BobAssume an “AND” test with perfect gamesOutputs must be low entropyIdea: Replay previous outputs when inputs repeat.
But, how can we be sure when inputs repeat Slide13
Doubly Exponential Bound
Idea: Alice and Bob both compute input matrix MWhere rows of M repeat, inputs must repeatReplay outputs on repeated rows
(0, 1)(1, 1) (1, 0)(0, 1)
(1, 0)
(0, 0)
(0, 1)
(1, 1)
(1, 1)
(0, 0)
(1, 0)
(0, 1)
(0, 1)
(0, 0)
(1, 1)
(1, 0)
(0, 1)
(1, 0)
(0, 0)
(1, 1)
(1, 1)
(1, 0)
(0, 0)
(0, 1)
(1, 0)
(1, 1) (0, 0)(0, 1)(1, 1)
(0, 0)(1, 0)(1, 1)
....
...
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MSlide14
Doubly Exponential Bound
Suppose that the Referee’s seed is n bitsRows of M are 2n+1
bits longThere are at most
distinct rows of M
So only need to play that many fair games
(0, 1)
(1, 1)
(1, 0)
(0, 1)
(1, 0)
(0, 0)
(0, 1)
(1, 1)
(1, 1)
(0, 0)
(1, 0)
(0, 1)
(0, 1)
(0, 0)
(1, 1)
(1, 0)
(0, 1)
(1, 0)
(0, 0)
(1, 1)(1, 1)(1, 0)(0, 0)
(0, 1)
(1, 0)(1, 1) (0, 0)(0, 1)(1, 1)
(0, 0)(1, 0)(1, 1)
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2
n+1Slide15
Exponential Bound
Consider CHSH testsMany existing protocols use theseGoal: exhibit a “cheating strategy” for Alice and Bob
Require that they only play an exponential number of games honestly
100010100
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
Test
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1111101010
0100001010
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0Slide16
Exponential Bound
Idea: Imagine rows as vectorsThe dimension of the vector space is only exponential (not doubly)How can we use this?
Only play honestly on rows of M that are linearly independent of previous rows
(0, 1)
(1, 1)
(1, 0)
(0, 1)
(1, 0)
(0, 0)
(0, 1)
(1, 1)
(1, 1)
(0, 0)
(1, 0)
(0, 1)
(0, 1)
(0, 0)
(1, 1)
(1, 0)
(0, 1)
(1, 0)
(0, 0)
(1, 1)
(1, 1)(1, 0)(0, 0)(0, 1)
(1, 0)
(1, 1) (0, 0)(0, 1)(1, 1)(0, 0)
(1, 0)(1, 1)
....
...
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MSlide17
Exponential Bound
What about linearly dependent rows?Their inputs are linear combinations of previous inputs
X =
and Y
=
Want A,B
s.t.
A+B = X
Λ
Y
=
Λ
Slide18
Exponential Bound
Idea: Can pre-compute
,
such that
=
Λ
Alice and Bob can do this
by playing 2
O(n)
games in secret
100010100
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
Test
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1111101010
0100001010
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0Slide19
Exponential Bound
We have: X =
and Y
=
.
So,
i
f
A =
and B =
Then,
A+B = X
Λ
Y
=
Λ
.
Slide20
Open Problems
Adaptive protocolsMore General TestsOther Games
100010100
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
Test
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1111101010
0100001010
0
1
1
0
1
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0
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1
0
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1
0
1
0