Random non-local games

1. Random non-local games Andris Ambainis, Artūrs Bačkurs, Kaspars Balodis, Dmitry Kravchenko, Juris Smotrovs, Madars Virza University of Latvia

2. Non-local games Bob Alice a b x y Referee Referee asks questions a, b to Alice, Bob; Alice and Bob reply by sending x, y; Alice, Bob win if a condition Pa, b(x, y) satisfied.

3. Example 1 [CHSH] Bob Alice a b x y Referee Winning conditions for Alice and Bob (a = 0 or b = 0)  x = y. (a = b = 1)  x  y.

4. Example 2 [Cleve et al., 04] Alice and Bob attempt to “prove” that they have a 2-coloring of a 5-cycle; Referee may ask one question about color of some vertex to each of them.

5. Example 2 Referee either: asks ith vertex to both Alice and Bob; they win if answers equal. Asks the ith vertex to Alice, (i+1)st to Bob, they win if answers different.

6. Non-local games in quantum world  Bob Alice Corresponds to shared random bits in the classical case. • Shared quantum state between Alice and Bob: • Does not allow them to communicate; • Allows to generate correlated random bits.

7. Example:CHSH game Bob Alice a b x y Referee Winning condition: • (a = 0 or b = 0) x = y. • (a = b = 1)  x  y. Winning probability: • 0.75 classically. • 0.85... quantumly. A simple way to verify quantum mechanics.

8. Example: 2-coloring game Winning probability: • classically. • quantumly. Alice and Bob claim to have a 2-coloring of n-cycle, n- odd; 2n pairs of questions by referee.

9. Random non-local games Bob Alice a b x y Referee • a, b  {1, 2, ..., N}; • x, y  {0, 1}; • Condition P(a, b, x, y) – random; Computer experiments: quantum winning probability larger than classical.

10. XOR games • For each (a, b), exactly one of x = y and x  y is winning outcome for Alice and Bob.

11. The main results • Let N be the number of possible questions to Alice and Bob. • Classical winning probability pcl satisfies • Quantum winning probability pq satisfies

12. Another interpretation • Value of the game = pwin – (1-pwin). • Quantum advantage: Corresponds to Bell inequality violation

13. Related work • Randomized constructions of non-local games/Bell inequalities with large quantum advantage. • Junge, Palazuelos, 2010: N questions, N answers, √N/log N advantage. • Regev, 2011: √N advantage. • Briet, Vidick, 2011: large advantage for XOR games with 3 players.

14. Differences • JP10, R10, BV11: • Goal: maximize quantum-classical gap; • Randomized constructions = tool to achieve this goal; • This work: • Goal: understand the power of quantum strategies in random games;

15. Methods: quantum Tsirelson’s theorem, 1980: • Alice’s strategy - vectors u1, ..., uN, ||u1|| = ... = ||uN|| = 1. • Bob’s strategy - vectors v1, ..., vN, ||v1|| = ... = ||vN|| = 1. • Quantum advantage

16. Random matrix question • What is the value of for a random 1 matrix A? Can be upper-bounded using ||A||=(2+o(1)) √N

17. Upperbound =

18. Upper bound theorem • Theorem For a random A, • Corollary The advantage achievable by a quantum strategy in a random XOR game is at most

19. Lower bound • There exists u: • There are many such u: a subspace of dimension f(n), for any f(n)=o(n). • Combine them to produce ui, vj:

20. Marčenko-Pastur law • Let A – random N·N 1 matrix. • W.h.p., all singular values are between 0 and (2o(1)) √N. • Theorem (Marčenko, Pastur, 1967) W.h.p., the fraction of singular values i  c √N is

21. Modified Marčenko-Pastur law • Let e1, e2, ...,eN be the standard basis. • Theorem With probability 1-O(1/N), the projection of ei to the subspace spanned by singular values j  c √N is

22. Classical results Let N be the number of possible questions to Alice and Bob. Theorem Classical winning probability pcl satisfies

23. Methods: classical • Alice’s strategy - numbers u1, ..., uN {-1, 1}. • Bob’s strategy - numbers v1, ..., vN {-1, 1}. • Classical advantage

24. Classical upper bound • Chernoff bounds + union bound: with probability 1-o(1).

25. Classical lower bound • Random 1 matrix A; • Operations: flip signs in all entries in one column or row; • Goal: maximize the sum of entries.

26. Greedy strategy • Choose u1, ..., uN one by one. k-1 rows that are already chosen 2 0 -2 ... 0 Choose the option which maximizes agreements of signs

27. Analysis 2 0 -2 ... 0 • On average, the best option agrees with fraction of signs. Choose the option which maximizes agreements of signs • If the column sum is 0, it always increases.

28. Rigorous proof 2 0 -2 ... 0 • Consider each column separately. Sum of values performs a biased random walk, moving away from 0 with probability in each step. Choose the option which maximizes agreements of signs • Expected distance from origin = 1.27... √N.

29. Conclusion • We studied random XOR games with n questions to Alice and Bob. • For both quantum and classical strategies, the best winning probability  ½. • Quantumly: • Classically:

30. Comparison Random XOR game: • Biggest gap for XOR games:

31. Open problems • We have What is the exact order? • Gaussian Aij? Different probability distributions? • Random games for other classes of non-local games?