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Robust Randomness Expansion Upper and Lower Bounds

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.

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Robust Randomness Expansion Upper and Lower Bounds

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  1. Robust Randomness ExpansionUpper and Lower Bounds Matthew Coudron, Thomas Vidick, Henry Yuen arXiv:1305.6626

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

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

  4. The motivating question Is it possible to test randomness? 1000101001111…..

  5. The motivating question Is it possible to test randomness? 1111111111111…..

  6. The motivating question Is it possible to test randomness? No, not possible! 1111111111111…..

  7. No-signaling offers a way…

  8. No-signaling offers a way… No-signaling constraint makes testing randomness possible!

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

  10. CHSH game x ϵ {0,1} y ϵ {0,1} Idea [EPR, Bell]: if the devices win the CHSH game with > 75% success probability, then their outputs must be randomized! aϵ {0,1} b ϵ {0,1} CHSH condition: a+b = x Λ y Classical win probability: 75% Quantum win probability: ~85%

  11. Certifying randomness via CHSH Devices play n rounds of the CHSH game[Colbeck]. 1 0

  12. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 1 0 0 0

  13. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10 01 0 0

  14. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10 01 01 00

  15. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100 011 01 00

  16. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100 011 011 001

  17. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 1001 0111 011 001

  18. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 1001 0111 0110 0010

  19. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10010101010101010 0111010110101010 01101010101111000 0010111110101011 Won ~85% of rounds?

  20. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10010101010101010 0111010110101010 01101010101111000 0010111110101011 Outputs have W(n) bits of certified min-entropy!

  21. Certifying randomness via CHSH Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certifyrandomness, but also expandit! Short random seed 1000101001 Long pseudorandom input 10010101010101010 0111010110101010 01101010101111000 0010111110101011 Outputs have W(n) bits of certified min-entropy!

  22. Certifying randomness via CHSH Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certifyrandomness, but also expandit! Short random seed 1000101001 Long pseudorandom input 10010101010101010 0111010110101010 State-of-the-art: Vazirani-Vidick protocol uses m bits of seed and produces 2O(m) certified random bits! [VV12] 01101010101111000 0010111110101011 Outputs have W(n) bits of certified min-entropy!

  23. 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

  24. What are the possibilities? Limits? • Doubly exponential expansion? • …infinite expansion? • Noise robustness?

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

  26. 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 Hmin( device outputs ) > g(m) c – completeness s – soundness g(m) - expansion

  27. 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 Hmin( device outputs ) > g(m) • Non-adaptive • Inputs to devices don’t depend on their outputs c – completeness s – soundness g(m) - expansion

  28. 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) XOR game: game win condition depends only on parity of players’ answers. non-signaling strategies: strictly more powerful than quantum strategies. *IMpossibility results

  29. How to prove upper bounds? Exhibit a cheating strategy for the devices, i.e. a strategy Scheat where Pr ( Passing protocol with Scheat) > s but Hmin ( device outputs ) < g(m)

  30. 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

  31. 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!

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

  33. An exp(exp(m)) upper bound Input Matrix Referee’s random seed (2m columns) Repeat answers whenever rows repeat!

  34. 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.

  35. 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

  36. 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)

  37. 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?

  38. 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!

  39. 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!

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