Random nonlocal games Andris Ambainis Artrs Bakurs Kaspars

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

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

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

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

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

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

Example: CHSH game Alice a x Bob b Referee y Winning condition: Winning probability: l (a = 0 or b = 0) x = y. l 0. 75 classically. l (a = b = 1) x y. l 0. 85. . . quantumly. A simple way to verify quantum mechanics.

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

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

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

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

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

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

Differences l JP 10, R 10, BV 11: Goal: maximize quantum-classical gap; l Randomized constructions = tool to achieve this goal; l l This work: l Goal: understand the power of quantum strategies in random games;

Methods: quantum Tsirelson’s theorem, 1980: l Alice’s strategy - vectors u 1, . . . , u. N, ||u 1|| =. . . = ||u. N|| = 1. l Bob’s strategy - vectors v 1, . . . , v. N, ||v 1|| =. . . = ||v. N|| = 1. l Quantum advantage

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

Upper bound =

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

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

Marčenko-Pastur law l l l 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

Modified Marčenko-Pastur law l l Let e 1, e 2, . . . , e. N 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

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

Methods: classical Alice’s strategy - numbers u 1, . . . , u. N {-1, 1}. l Bob’s strategy - numbers v 1, . . . , v. N {-1, 1}. l Classical advantage l

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

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

Greedy strategy l Choose u 1, . . . , u. N one by one. 1 -1. . . 1 1 -1. . . -1 . . . 1. . -1 2 0 -2 . . . -1 1 1 . . . 1 1 -1 -1 . . . -1 k-1 rows that are already chosen 0 Choose the option which maximizes agreements of signs

Analysis 2 0 -2 . . . 0 -1 1 1 . . . 1 1 -1 -1 . . . -1 Choose the option which maximizes agreements of signs l On average, the best option agrees with fraction of signs. l If the column sum is 0, it always increases.

Rigorous proof 2 0 -2 . . . 0 -1 1 1 . . . 1 1 -1 -1 . . . -1 Choose the option which maximizes agreements of signs l Consider each column separately. Sum of values performs a biased random walk, moving away from 0 with probability in each step. l Expected distance from origin = 1. 27. . . √N.

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

Comparison l Random XOR game: l Biggest gap for XOR games:

Open problems 1. 2. 3. We have What is the exact order? Gaussian Aij? Different probability distributions? Random games for other classes of non-local games?