robust randomness expansion upper and lower bounds matthew coudron, thomas vidick, henry yuen...
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![Page 1: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/1.jpg)
Robust Randomness Expansion
Upper and Lower Bounds
Matthew Coudron, Thomas Vidick, Henry Yuen
arXiv:1305.6626
![Page 2: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/2.jpg)
The motivating question
Is it possible to test randomness?
![Page 3: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/3.jpg)
The motivating question
Is it possible to test randomness?
![Page 4: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/4.jpg)
The motivating question
Is it possible to test randomness?
1000101001111…..
![Page 5: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/5.jpg)
The motivating question
Is it possible to test randomness?
1111111111111…..
![Page 6: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/6.jpg)
The motivating question
Is it possible to test randomness?
1111111111111…..
No, not possible!
![Page 7: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/7.jpg)
No-signaling offers a way…
![Page 8: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/8.jpg)
No-signaling offers a way…
No-signaling constraint makes testing randomness possible!
![Page 9: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/9.jpg)
CHSH gamex ϵ {0,1}
y ϵ {0,1}
a ϵ {0,1}
b ϵ {0,1}
CHSH condition: a+b = x Λ y
Classical win probability: 75%
Quantum win probability: ~85%
![Page 10: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/10.jpg)
CHSH gamex ϵ {0,1}
y ϵ {0,1}
a ϵ {0,1}
b ϵ {0,1}
CHSH condition: a+b = x Λ y
Classical win probability: 75%
Quantum win probability: ~85%
Idea [EPR, Bell]: if the devices win the CHSH game
with > 75% success probability, then their outputs
must be randomized!
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Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1 0
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Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1 0
0 0
![Page 13: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/13.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10 01
0 0
![Page 14: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/14.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10 01
01 00
![Page 15: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/15.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
100 011
01 00
![Page 16: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/16.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
100 011
011 001
![Page 17: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/17.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1001 0111
011 001
![Page 18: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/18.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1001 0111
0110 0010
![Page 19: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/19.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10010101010101010
0111010110101010
01101010101111000
0010111110101011
Won ~85% of rounds?
![Page 20: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/20.jpg)
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10010101010101010
0111010110101010
01101010101111000
0010111110101011
Outputs have (W n) bits of certified min-entropy!
![Page 21: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/21.jpg)
Certifying randomness via CHSH
10010101010101010
0111010110101010
01101010101111000
0010111110101011
Outputs have (W n) bits of certified min-entropy!
Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it!
1000101001
Short random seed
Long pseudorandom input
![Page 22: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/22.jpg)
Certifying randomness via CHSH
10010101010101010
0111010110101010
01101010101111000
0010111110101011
Outputs have (W n) bits of certified min-entropy!
Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it!
1000101001
Short random seed
Long pseudorandom input
State-of-the-art: Vazirani-Vidick protocol uses m bits of seed and produces 2O(m) certified
random bits! [VV12]
![Page 23: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/23.jpg)
How do we measure randomness?
We use min-entropy. For a random variable X,
Hmin (X) := min log 1/Pr(X = x)
Why min-entropy? It characterizes the amount of uniformly random bits that one can extract from a random source X!
x
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What are the possibilities? Limits?
• Doubly exponential expansion?
• …infinite expansion?
• Noise robustness?
![Page 25: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/25.jpg)
Our results
• First upper bounds for non-adaptive randomness expansion
• Constructions of noise-robust protocols
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The modelRandomness amplifier is an interactive protocol between a classical referee and 2 non-signaling devices.
• Randomness efficiency• Referee uses m random bits to sample inputs to devices
• Completeness• There exists an ideal strategy that passes the protocol
with probability > c
• Soundness• For all strategies S, if the devices using S, pass with
probability > s, then Hmin( device outputs ) > g(m)
c – completeness s – soundness g(m) - expansion
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The modelRandomness amplifier is an interactive protocol between a classical referee and 2 non-signaling devices.
• Randomness efficiency• Referee uses m random bits to sample inputs to devices
• Completeness• There exists an ideal strategy that passes the protocol
with probability > c
• Soundness• For all strategies S, if the devices using S, pass with
probability > s, then Hmin( device outputs ) > g(m)
• Non-adaptive• Inputs to devices don’t depend on their outputs
c – completeness s – soundness g(m) - expansion
![Page 28: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/28.jpg)
Upper bounds*
1. Noise-robust randomness amplifiers- g(m) < exp(exp(m))
2. Randomness amplifiers using XOR games and devices have non-signaling power
- g(m) < exp(m)
*IMpossibility results
XOR game: game win condition depends only on parity of players’ answers.
non-signaling strategies: strictly more powerful than quantum strategies.
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How to prove upper bounds?
Exhibit a cheating strategy for the devices,
i.e. a strategy Scheat where
Pr ( Passing protocol with Scheat ) > sbut
Hmin ( device outputs ) < g(m)
![Page 30: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/30.jpg)
An exp(exp(m)) upper bound
• Our main doubly-exp upper bound applies to non-adaptive, noise-robust randomness amplifiers
• A proof for a simplified setting:• Protocols based on perfect games (e.g.
Magic Square)• Referee check devices won every round
![Page 31: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/31.jpg)
An exp(exp(m)) upper bound
Intuition: after exp(exp(m)) rounds, inputs to the devices will start repeating in predictable ways…
Independently of referee’s private randomness!
![Page 32: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/32.jpg)
An exp(exp(m)) upper bound
Input Matrix
0000 0001 …. 1110 1111(1, 0) (0, 1) (1,0) (1,1)
(1,1) (0,1) (1,1) (1,1)
(0,0) (0,0) (0,0) (1,0)
…
(1,0) (0,1) (1,0) (1,1)
Referee’s random seed (2m columns)
Input to devices
in round i
After exp(exp(m)) rounds, rows must
start repeating
![Page 33: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/33.jpg)
An exp(exp(m)) upper bound
Input Matrix
0000 0001 …. 1110 1111(1, 0) (0, 1) (1,0) (1,1)
(1,1) (0,1) (1,1) (1,1)
(0,0) (0,0) (0,0) (1,0)
…
(1,0) (0,1) (1,0) (1,1)
Referee’s random seed (2m columns)
Repeat answers
whenever rows
repeat!
![Page 34: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/34.jpg)
An exp(exp(m)) upper bound
• Strategy Scheat
• Play “honestly” in round i when row i of Input Matrix is new
• If row i is a repeat of row j for some j < i, repeat answers from round j.
• Claim. Devices produce at most exp(exp(m)) bits of randomness, but pass protocol with probability 1.
![Page 35: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/35.jpg)
Generalizing the upper bound
• What if the referee is more clever? • Checks for obvious answer repetitions• Uses a non-perfect game, like odd-cycle
game or CHSH*• Still have exp(exp(m)) upper bound!
• Requirement for noise robustness gives devices freedom to cheat!
* For quantum players
![Page 36: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/36.jpg)
An exponential upper bound
• Cheating strategies that take advantage of the game structure
• XOR-game protocols• XOR game: f(x + y)• Devices can employ full non-signaling
strategies (i.e. super-quantum strategies)
• Referee checks devices won every round• g(m) < exp(m)
![Page 37: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/37.jpg)
Open problems
• Better upper bounds?–More elaborate cheating strategies?– Show g(m) < exp(m) always?
• Better lower bounds?–Match the doubly exponential upper
bound?
• Adaptive protocols with infinite expansion?
![Page 38: Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626](https://reader036.vdocuments.mx/reader036/viewer/2022070411/56649c9a5503460f94957065/html5/thumbnails/38.jpg)
Open problems
• Better upper bounds?–More elaborate cheating strategies?– Show g(m) < exp(m) always?
• Better lower bounds?–Match the doubly exponential upper
bound?
• Adaptive protocols with infinite expansion?
Thanks!
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