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Robust Randomness Expansion Upper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:

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The motivating question Is it possible to test randomness?

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The motivating question Is it possible to test randomness?

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The motivating question Is it possible to test randomness? …..

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The motivating question Is it possible to test randomness? …..

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The motivating question Is it possible to test randomness? ….. No, not possible!

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No-signaling offers a way…

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No-signaling constraint makes testing randomness possible!

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CHSH game x {0,1}y {0,1} a {0,1}b {0,1} CHSH condition: a+b = x Λ y Classical win probability: 75%Quantum win probability: ~85%

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CHSH game x {0,1}y {0,1} a {0,1}b {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 CHSH Devices play n rounds of the CHSH game [Colbeck]. 10

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck] Won ~85% of rounds?

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Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck] Outputs have n) bits of certified min-entropy!

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Certifying randomness via CHSH Outputs have n) bits of certified min-entropy! Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it! Short random seed Long pseudorandom input

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Certifying randomness via CHSH Outputs have n) bits of certified min-entropy! Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it! Short random seed Long pseudorandom input State-of-the-art: Vazirani-Vidick protocol uses m bits of seed and produces 2 O(m) certified random bits! [VV12]

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How do we measure randomness? We use min-entropy. For a random variable X, H min (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?

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Our results First upper bounds for non-adaptive randomness expansion Constructions of noise-robust protocols

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The model Randomness 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 H min ( device outputs ) > g(m) c – completenesss – soundnessg(m) - expansion

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The model Randomness 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 H min ( device outputs ) > g(m) Non-adaptive Inputs to devices don’t depend on their outputs c – completenesss – soundnessg(m) - expansion

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Lower bounds* Noise-robust randomness amplifier Exponential expansion (i.e. g(m) = 2 O(m) ) Simplified protocol Works with any randomness generating game (not just CHSH) *possibility results

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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 S cheat where Pr ( Passing protocol with S cheat ) > s but H min ( device outputs ) < g(m)

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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

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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!

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An exp(exp(m)) upper bound Input Matrix … (1, 0)(0, 1)(1,0)(1,1) (0,1)(1,1) (0,0) (1,0) … (0,1)(1,0)(1,1) Referee’s random seed (2 m columns) Input to devices in round i After exp(exp(m)) rounds, rows must start repeating

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An exp(exp(m)) upper bound Input Matrix … (1, 0)(0, 1)(1,0)(1,1) (0,1)(1,1) (0,0) (1,0) … (0,1)(1,0)(1,1) Referee’s random seed (2 m columns) Repeat answers whenever rows repeat!

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An exp(exp(m)) upper bound Strategy S cheat 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.

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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

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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)

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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?

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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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