1. Robust Randomness Expansion Upper 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? 1111111111111….. No, not possible!

7. No-signaling offers a way…

8. No-signaling constraint makes testing randomness possible!

9. 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%

10. 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!

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

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

13. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 1001 00

14. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 1001 00

15. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100011 0100

16. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100011 001

17. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10010111 011001

18. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 10010111 01100010

19. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100101010101010100111010110101010 011010101011110000010111110101011 Won ~85% of rounds?

20. Certifying randomness via CHSH Devices play n rounds of the CHSH game [Colbeck]. 100101010101010100111010110101010 011010101011110000010111110101011 Outputs have  n) bits of certified min-entropy!

21. Certifying randomness via CHSH 100101010101010100111010110101010 011010101011110000010111110101011 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! 1000101001 Short random seed Long pseudorandom input

22. Certifying randomness via CHSH 100101010101010100111010110101010 011010101011110000010111110101011 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! 1000101001 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]

23. 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

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

28. 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

29. 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.

30. 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)

31. 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

32. 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!

33. An exp(exp(m)) upper bound Input Matrix 00000001….11101111 (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

34. An exp(exp(m)) upper bound Input Matrix 00000001….11101111 (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!

35. 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.

36. 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

37. 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)

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?

39. 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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