Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen...
-
Upload
rodrigo-warn -
Category
Documents
-
view
212 -
download
0
Transcript of Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen...
Robust Randomness Expansion
Upper and Lower Bounds
Matthew Coudron, Thomas Vidick, Henry Yuen
arXiv:1305.6626
The motivating question
Is it possible to test randomness?
The motivating question
Is it possible to test randomness?
The motivating question
Is it possible to test randomness?
1000101001111…..
The motivating question
Is it possible to test randomness?
1111111111111…..
The motivating question
Is it possible to test randomness?
1111111111111…..
No, not possible!
No-signaling offers a way…
No-signaling offers a way…
No-signaling constraint makes testing randomness possible!
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%
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!
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1 0
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1 0
0 0
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10 01
0 0
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10 01
01 00
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
100 011
01 00
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
100 011
011 001
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1001 0111
011 001
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
1001 0111
0110 0010
Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].
10010101010101010
0111010110101010
01101010101111000
0010111110101011
Won ~85% of rounds?
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!
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
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]
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
What are the possibilities? Limits?
• Doubly exponential expansion?
• …infinite expansion?
• Noise robustness?
Our results
• First upper bounds for non-adaptive randomness expansion
• Constructions of noise-robust protocols
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
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
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.
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)
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
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!
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
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!
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.
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
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)
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?
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!
Advertisement
• I’m an organizer of the Algorithms & Complexity seminar this term.
• If you’re in the Boston area, and want to give a talk at MIT, let me know!